{"id":68,"date":"2023-09-17T05:59:03","date_gmt":"2023-09-17T05:59:03","guid":{"rendered":"https:\/\/mathority.org\/id\/cara-mendiagonalisasi-matriks-yang-dapat-didiagonalisasi-latihan-diagonalisasi-matriks-2x2-3x3-4x4-diselesaikan-langkah-demi-langkah\/"},"modified":"2023-09-17T05:59:03","modified_gmt":"2023-09-17T05:59:03","slug":"cara-mendiagonalisasi-matriks-yang-dapat-didiagonalisasi-latihan-diagonalisasi-matriks-2x2-3x3-4x4-diselesaikan-langkah-demi-langkah","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/cara-mendiagonalisasi-matriks-yang-dapat-didiagonalisasi-latihan-diagonalisasi-matriks-2x2-3x3-4x4-diselesaikan-langkah-demi-langkah\/","title":{"rendered":"Cara mendiagonalisasi matriks"},"content":{"rendered":"<p>Di halaman ini Anda akan menemukan segala sesuatu tentang matriks yang dapat didiagonalisasi: apa itu matriks, kapan matriks tersebut dapat didiagonalisasi dan kapan tidak dapat didiagonalisasi, metode untuk mendiagonalisasi matriks, penerapan dan sifat-sifat matriks tertentu, dll. Dan Anda bahkan memiliki beberapa latihan yang diselesaikan selangkah demi selangkah sehingga Anda dapat berlatih dan memahami dengan sempurna bagaimana latihan tersebut didiagonalisasi. Terakhir, kita juga mempelajari cara melakukan diagonalisasi matriks dengan program komputer MATLAB, karena program ini sangat sering digunakan.<\/p>\n<h2 class=\"wp-block-heading\"> Apa yang dimaksud dengan matriks yang dapat didiagonalisasi?<\/h2>\n<p> Seperti yang akan kita lihat di bawah, mendiagonalisasi suatu matriks sangat berguna dalam bidang aljabar linier. Makanya banyak yang bertanya\u2026 apa itu diagonalisasi matriks? Nah, pengertian matriks yang dapat didiagonalisasi adalah: <\/p>\n<div style=\"background-color:#dff6ff;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px\" class=\"has-background\">\n<p style=\"text-align:left\"> <strong>Matriks yang dapat didiagonalisasi<\/strong> adalah matriks persegi yang dapat diubah menjadi matriks diagonal, yaitu matriks yang berisi angka nol kecuali pada diagonal utama. Diagonalisasi matriks dipecah sebagai berikut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4ab9c489a73de0bde368d8a7f7bd7151_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A = PDP^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"97\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> Atau setara,<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-54fc1390aaa9437bf9813fc64b600919_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D = P^{-1}AP\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"98\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> Emas<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> adalah matriks yang akan didiagonalisasi,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> adalah matriks yang kolomnya merupakan vektor eigen (atau vektor eigen).<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> ,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-073aeddfae03d7bea03931e1cb3505f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"31\" style=\"vertical-align: 0px;\"><\/p>\n<p> matriks inversnya dan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b9ef1bbd23fd1b198de883813285620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> adalah matriks diagonal yang dibentuk oleh nilai eigen (atau nilai eigen) dari<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> .<\/p>\n<\/div>\n<p> Matriks<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> bertindak sebagai matriks perubahan basis, jadi sebenarnya dengan rumus ini kita mengubah basis menjadi matriks<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> , sehingga matriks tersebut menjadi matriks diagonal (<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b9ef1bbd23fd1b198de883813285620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> ) di pangkalan baru.<\/p>\n<p> Oleh karena itu, matriks<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> dan matriks<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b9ef1bbd23fd1b198de883813285620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> Mereka adalah matriks yang serupa. Dan jelas,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> Ini adalah matriks reguler atau non-degenerasi.<\/p>\n<h2 class=\"wp-block-heading\"> Kapankah suatu matriks dapat didiagonalisasi?<\/h2>\n<p> Tidak semua matriks dapat didiagonalisasi; hanya matriks yang memenuhi karakteristik tertentu yang dapat didiagonalisasi. Anda dapat mengetahui apakah suatu matriks dapat didiagonalisasi dengan berbagai cara:<\/p>\n<ul>\n<li> Suatu matriks persegi berorde <em>n<\/em> dapat didiagonalisasi jika mempunyai <em>n<\/em> <span style=\"color:#1976d2;\"><strong>vektor eigen (atau vektor eigen) yang bebas linier<\/strong><\/span> , atau dengan kata lain, jika vektor-vektor tersebut membentuk basis. Hal ini karena matriks\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> , yang digunakan untuk mendiagonalisasi suatu matriks, dibentuk oleh vektor eigen matriks tersebut. Untuk mengetahui apakah vektor eigennya LI, cukup determinan matriksnya<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> berbeda dengan 0 yang berarti matriks mempunyai rank maksimum.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-389610a3ba8bf2db8af148a3f5c13e5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{si} \\quad \\text{det}(P)\\neq 0 \\ \\longrightarrow \\ \\text{matriz diagonalizable}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"331\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Sifat nilai eigen dan vektor eigen adalah bahwa vektor eigen dari nilai eigen yang berbeda adalah bebas linier. Oleh karena itu, jika <span style=\"color:#1976d2;\"><strong>semua nilai eigen matriks adalah unik, maka<\/strong><\/span> matriks tersebut dapat didiagonalisasi.<\/li>\n<\/ul>\n<ul>\n<li> Cara lain untuk menentukan apakah suatu matriks dapat ditampung dalam matriks diagonal adalah dengan menggunakan multiplisitas aljabar dan geometri. Multiplisitas aljabar adalah berapa kali suatu nilai eigen (atau nilai eigen) diulang, dan multiplisitas geometri adalah dimensi inti (atau inti) matriks dengan mengurangkan nilai eigen pada diagonal utamanya. Jadi, jika untuk setiap nilai eigen <span style=\"color:#1976d2;\"><strong>multiplisitas aljabar sama dengan multiplisitas geometri<\/strong><\/span> , matriks tersebut dapat didiagonalisasi. <\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cee403ec4a2cac29cda0bf950fcc143b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha_\\lambda = \\text{multiplicidad algebraica} = \\text{multiplicidad del valor propio}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"483\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-968bde68480ba0b85f5179a1a794bfec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m_\\lambda = \\text{multiplicidad geom\\'etrica} = \\text{dim } Ker(A-\\lambda I) = n -rg(A-\\lambda I)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"541\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8b7ecdb0203a83bf48683c551df7418a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha_\\lambda \\geq m_\\lambda \\geq 1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"100\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-511e6243e12d5227417f12bb1ef29330_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{si} \\quad \\alpha_\\lambda = m_\\lambda \\quad \\forall \\lambda \\ \\longrightarrow \\ \\text{matriz diagonalizable}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"353\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<ul>\n<li> Terakhir, ada teorema, teorema spektral, yang menjamin diagonalisasi matriks simetris dengan bilangan real. Dengan kata lain, <span style=\"color:#1976d2;\"><strong>setiap matriks real dan simetris dapat didiagonalisasi<\/strong><\/span> .<\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"> Cara mendiagonalisasi matriks<\/h2>\n<p> Prosedur untuk mendiagonalisasi suatu matriks didasarkan pada pencarian nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) suatu matriks. Inilah mengapa penting bagi Anda untuk menguasai <a href=\"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/\">cara menghitung nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) dari matriks apa pun<\/a> . Anda dapat mengingat cara melakukannya dengan mengeklik tautannya, di mana kami menjelaskan langkah demi langkah cara menemukannya dan beberapa trik yang membuat penghitungan menjadi lebih mudah. Selain itu, Anda juga akan menemukan latihan terpecahkan untuk dipraktikkan.<\/p>\n<p> Dengan metode berikut, Anda dapat mendiagonalisasi matriks dengan dimensi apa pun: 2&#215;2, 3&#215;3, 4&#215;4, dst. Langkah-langkah yang harus diikuti untuk mendiagonalisasi suatu matriks adalah:<\/p>\n<ol style=\"color:#1976d2; font-weight: bold;>\n<li><span style=\" color:#262626;font-weight:=\"\" normal;\"=\"\">\n<li style=\"margin-bottom:23px\"><span style=\"color:#262626;font-weight: normal;\">Dapatkan nilai eigen (atau nilai eigen) dari matriks.<\/span><\/li>\n<li style=\"margin-bottom:23px\"> <span style=\"color:#262626;font-weight: normal;\">Hitung vektor eigen yang terkait dengan setiap nilai eigen.<\/span><\/li>\n<li style=\"margin-bottom:23px\"> <span style=\"color:#262626;font-weight: normal;\">Bangun matriksnya\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p><\/span> , yang kolomnya merupakan vektor eigen dari matriks yang akan didiagonalisasi.<\/li>\n<li style=\"margin-bottom:23px\"> <span style=\"color:#262626;font-weight: normal;\">Periksa apakah matriks dapat didiagonalisasi (harus memenuhi salah satu syarat yang dijelaskan pada bagian sebelumnya).<\/span><\/li>\n<li> <span style=\"color:#262626;font-weight: normal;\">Buatlah matriks diagonal\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b9ef1bbd23fd1b198de883813285620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p><\/span> , yang elemennya semuanya 0 kecuali yang ada pada diagonal utama, yang merupakan nilai eigen yang ditemukan pada langkah 1.<\/li>\n<\/ol>\n<p class=\"has-background\" style=\"background-color:#fffde7\"> <strong>Peringatan:<\/strong> Vektor eigen dari matriks<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> dapat ditempatkan dalam urutan apa pun, kecuali nilai eigen matriks diagonal<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b9ef1bbd23fd1b198de883813285620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> Mereka harus ditempatkan dalam urutan yang sama. Misalnya nilai eigen pertama dari matriks diagonal<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b9ef1bbd23fd1b198de883813285620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> haruslah vektor yang sesuai dengan vektor eigen kolom pertama matriks<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> .<\/p>\n<p> Di bawah ini beberapa latihan diagonalisasi matriks langkah demi langkah yang dapat Anda praktikkan.<\/p>\n<h2 class=\"wp-block-heading\"> Latihan diagonalisasi matriks diselesaikan<\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Diagonalisasikan matriks persegi berdimensi 2\u00d72 berikut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-13b9f5c8b5a381c9661aa4ee2e0b7b63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}2&amp;2\\\\[1.1ex] 1&amp;3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"95\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama-tama kita harus menentukan nilai eigen matriks A. Oleh karena itu, kita menghitung persamaan karakteristik dengan menyelesaikan determinan berikut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f652aa2ef8cd55100970fef7fbf30e60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}2- \\lambda &amp;2\\\\[1.1ex] 1&amp;3-\\lambda \\end{vmatrix} = \\lambda^2-5\\lambda +4\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"339\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang mari kita hitung akar-akar polinomial karakteristik:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c53bbe295e0f77d1cdaa183e9341567d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda^2-5\\lambda +4=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda = 4 \\\\[2ex] \\lambda = 1 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"231\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Setelah nilai eigen diperoleh, kami menghitung vektor eigen yang terkait dengan masing-masing nilai eigen. Pertama, vektor eigen yang sesuai dengan nilai eigen 1: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-10506efea4c355e8449378bc3a1948a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-83d7b7d31a262f6e0844a0a9f5098e11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}1&amp;2\\\\[1.1ex] 1&amp;2\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\end{pmatrix} =}\\begin{pmatrix}0 \\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"156\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9ea3248973afa32a42f87b20e0c5ddc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} x+2y = 0 \\\\[2ex] x+2y = 0\\end{array}\\right\\} \\longrightarrow \\ x=-2y\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-73435e1b8c9d689ec17255f087e978f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}-2 \\\\[1.1ex] 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"79\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kemudian kita menghitung vektor eigen yang terkait dengan nilai eigen 4: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0545c0847763140ccc62a58cf4207c6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-4I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d9c6b33d8fad6974d366ce088800b92a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}-2&amp;2\\\\[1.1ex] 1&amp;-1\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\end{pmatrix} =}\\begin{pmatrix}0 \\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"183\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d68533e14c844cf5bd4ee1965533ee6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -2x+2y = 0 \\\\[2ex] x-y = 0\\end{array}\\right\\} \\longrightarrow \\ y=x\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0f3cac5769795f1730fcbf118fdfbbc3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"66\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami membangun matriks<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> , dibentuk oleh vektor eigen matriks:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-46dde85eb30324e4dfec09cbb802853e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  P = \\begin{pmatrix}-2&amp;1 \\\\[1.1ex] 1&amp;1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"109\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Karena semua nilai eigennya berbeda, maka matriks A dapat didiagonalisasi. Jadi matriks diagonal yang bersesuaian adalah matriks yang mempunyai nilai eigen pada diagonal utama:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1e0329677969153d43ce741754dc6924_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle D= \\begin{pmatrix}1&amp;0\\\\[1.1ex] 0&amp;4\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"97\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ingatlah bahwa nilai eigen harus ditempatkan dalam urutan yang sama dengan penempatan vektor eigen dalam matriks<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> .<\/p>\n<p class=\"has-text-align-left\"> Kesimpulannya, matriks perubahan basis dan matriks yang didiagonalisasi adalah: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7ea54fe9e11d849c2896cc312df404ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle P = \\begin{pmatrix}-2&amp;1 \\\\[1.1ex] 1&amp;1 \\end{pmatrix} \\qquad D= \\begin{pmatrix}1&amp;0\\\\[1.1ex] 0&amp;4\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"248\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 2<\/h3>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<p> Diagonalisasikan matriks persegi berorde 2 berikut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f61af0f4b152be75cc74b7733b2de076_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}3&amp;4\\\\[1.1ex] -1&amp;-2\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"122\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama-tama kita harus menentukan nilai eigen matriks A. Oleh karena itu, kita menghitung persamaan karakteristik dengan menyelesaikan determinan berikut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-31024cc955652299f8933e082f934f15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}3- \\lambda &amp;4\\\\[1.1ex] -1&amp;-2-\\lambda \\end{vmatrix} = \\lambda^2-\\lambda -2\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"343\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang mari kita hitung akar-akar polinomial karakteristik:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-341b01a85529d26a506ebc9336221dca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda^2-\\lambda -2=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda = -1 \\\\[2ex] \\lambda = 2 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"235\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Setelah nilai eigen diperoleh, kami menghitung vektor eigen yang terkait dengan masing-masing nilai eigen. Pertama, vektor eigen yang sesuai dengan nilai eigen -1: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-76cc8bb12c3b49d4964b2b3f661677ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A+I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c2728e62bfb96bb9106b0f7791ba9c5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}4&amp;4\\\\[1.1ex] -1&amp;-1\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\end{pmatrix} =}\\begin{pmatrix}0 \\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"183\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0515ba12f6ad51cc35cc785697498b78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 4x+4y = 0 \\\\[2ex] -x-y = 0\\end{array}\\right\\} \\longrightarrow \\ x=-y\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0059538893e6c8439792228733f803de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}-1 \\\\[1.1ex] 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"79\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kemudian kita menghitung vektor eigen yang terkait dengan nilai eigen 2: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c6944f71d79a33d4789affbc82db4c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-2I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aff94f6ac5c08a408abcb42f4262ac0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}1&amp;4\\\\[1.1ex] -1&amp;-4\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\end{pmatrix} =}\\begin{pmatrix}0 \\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"183\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2930b2bf0ef86ea8be216bafe5c3aa32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} x+4y = 0 \\\\[2ex] -x-4y = 0\\end{array}\\right\\} \\longrightarrow \\ x=-4y\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"230\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede8a2a2803fc807b34db04326f5e1cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}-4 \\\\[1.1ex] 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"79\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami membangun matriks<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> , dibentuk oleh vektor eigen matriks:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-990ff35382717a4644e33e8630777237_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  P = \\begin{pmatrix}-1&amp;-4 \\\\[1.1ex] 1&amp;1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"123\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Karena semua nilai eigen berbeda satu sama lain, matriks A dapat didiagonalisasi. Jadi matriks diagonal yang bersesuaian adalah matriks yang memuat nilai eigen pada diagonal utama:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cb2a5e7884d62f8ed609465e289fa70e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle D= \\begin{pmatrix}-1&amp;0\\\\[1.1ex] 0&amp;2\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"110\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ingatlah bahwa nilai eigen harus ditempatkan dalam urutan yang sama dengan penempatan vektor eigen dalam matriks<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> .<\/p>\n<p class=\"has-text-align-left\"> Kesimpulannya, matriks perubahan basis dan matriks yang didiagonalisasi adalah: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-60358333f16516ac0d64d12891ef6ea5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle P = \\begin{pmatrix}-1&amp;-4 \\\\[1.1ex] 1&amp;1\\end{pmatrix} \\qquad D= \\begin{pmatrix}-1&amp;0\\\\[1.1ex] 0&amp;2\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"276\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 3<\/h3>\n<p> Diagonalisasikan matriks persegi berdimensi 3\u00d73 berikut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8fc8797c8c0354ff540e340b82cb9258_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}2&amp;0&amp;2\\\\[1.1ex] -1&amp;2&amp;1\\\\[1.1ex] 0&amp;1&amp;4\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"136\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Langkah pertama adalah mencari nilai eigen matriks A. Oleh karena itu, kita menghitung persamaan karakteristik dengan menyelesaikan determinan matriks berikut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-280aeb93bd229f34fe255f368390ae6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}2-\\lambda&amp;0&amp;2\\\\[1.1ex] -1&amp;2-\\lambda&amp;1\\\\[1.1ex] 0&amp;1&amp;4-\\lambda \\end{vmatrix} = -\\lambda^3+8\\lambda^2-19\\lambda+12\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"476\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang kita harus menghitung akar-akar polinomial karakteristik. Karena merupakan polinomial derajat ketiga, kami menerapkan aturan Ruffini:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9fc16d8c9420ece9152119b48f249df9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{array}{r|rrrr} &amp; -1&amp;8&amp;-19&amp; 12 \\\\[2ex] 1 &amp; &amp; -1&amp;7&amp;-12 \\\\ \\hline &amp;-1\\vphantom{\\Bigl)}&amp;7&amp;-12&amp;0 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"93\" width=\"199\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kemudian kita menemukan akar polinomial yang diperoleh: <\/p>\n<p class=\"has-text-align-center\">\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e07297a9da80525b94e7af1914f403be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -\\lambda^2+7\\lambda -12=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda = 3 \\\\[2ex] \\lambda = 4 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"252\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi nilai eigen matriksnya adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-798eb40221e94ae6f384d824bcc76998_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda=1 \\qquad \\lambda =3 \\qquad \\lambda = 4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"200\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Setelah nilai eigen ditemukan, kami menghitung vektor eigen yang terkait dengan masing-masing nilai tersebut. Pertama, vektor eigen yang sesuai dengan nilai eigen 1: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-10506efea4c355e8449378bc3a1948a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c50b6b424d6465f208981f3f89213bb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}1&amp;0&amp;2\\\\[1.1ex] -1&amp;1&amp;1\\\\[1.1ex] 0&amp;1&amp;3\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0 \\\\[1.1ex] 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"202\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8c810793db36ac827b71d01324760cee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} x+2z = 0 \\\\[2ex] -x+y+z = 0\\\\[2ex] y+3z = 0\\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l}x=-2z \\\\[2ex] y = -3z \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"260\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88fc97c4f3a0e5a6d79978e154230e22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}-2 \\\\[1.1ex] -3 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kemudian kita menghitung vektor eigen yang terkait dengan nilai eigen 3: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-50e802072a0f6e2942bc873d6a466909_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-3I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-05bcbb328be85066bb142c990bbfad99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}-1&amp;0&amp;2\\\\[1.1ex] -1&amp;-1&amp;1\\\\[1.1ex] 0&amp;1&amp;1\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0 \\\\[1.1ex] 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-656b90b758fe5ab6178efdfcbef399ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -x+2z = 0 \\\\[2ex] -x-y+z = 0\\\\[2ex] y+z = 0\\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l}x=2z \\\\[2ex] y = -z \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"250\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-33033f69510447ef3684a67e835bd578_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}2 \\\\[1.1ex] -1 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan terakhir, kami menghitung vektor eigen yang terkait dengan nilai eigen 4: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0545c0847763140ccc62a58cf4207c6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-4I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1f455bc39f72a9b8141ba714bd72a0e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}-2&amp;0&amp;2\\\\[1.1ex] -1&amp;-2&amp;1\\\\[1.1ex] 0&amp;1&amp;0\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0 \\\\[1.1ex] 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dd0ebc259ff2665ef3c4c3a3b1692e2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -2x+2z = 0 \\\\[2ex] -x-2y+z = 0\\\\[2ex] y = 0\\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l}x=z \\\\[2ex] y = 0 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"246\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-308b2f0f597fcc084d8d06d6c45fd3e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] 0 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami membangun matriks<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> , dibentuk oleh vektor eigen matriks:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f1f57ccbb391403b5e4af625900516cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  P = \\begin{pmatrix}-2&amp;2&amp;1 \\\\[1.1ex] -3&amp;-1&amp;0 \\\\[1.1ex] 1&amp;1&amp;1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"150\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Karena semua nilai eigen berbeda satu sama lain, matriks A dapat didiagonalisasi. Jadi matriks diagonal yang bersesuaian adalah matriks yang mempunyai nilai eigen pada diagonal utamanya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e88d2a690d31a9ca772d185078f69d3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle D= \\begin{pmatrix}1&amp;0&amp;0\\\\[1.1ex] 0&amp;3&amp;0 \\\\[1.1ex] 0&amp;0&amp;4\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"124\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ingatlah bahwa nilai eigen harus ditempatkan dalam urutan yang sama dengan penempatan vektor eigen dalam matriks<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> .<\/p>\n<p class=\"has-text-align-left\"> Singkatnya, matriks perubahan basis dan matriks yang didiagonalisasi adalah: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d6fdff19d2d1e3f58ba1898dc456711d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle P = \\begin{pmatrix}-2&amp;2&amp;1 \\\\[1.1ex] -3&amp;-1&amp;0 \\\\[1.1ex] 1&amp;1&amp;1\\end{pmatrix} \\qquad D= \\begin{pmatrix}1&amp;0&amp;0\\\\[1.1ex] 0&amp;3&amp;0 \\\\[1.1ex] 0&amp;0&amp;4\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"318\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 4<\/h3>\n<p> Diagonalisasikan, jika memungkinkan, matriks persegi berorde 3 berikut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-628e7e12a0d8ccde5bb1fb2626663910_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}-1&amp;3&amp;1\\\\[1.1ex] 0&amp;2&amp;0\\\\[1.1ex] 3&amp;-1&amp;1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"150\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Langkah pertama adalah mencari nilai eigen matriks A. Oleh karena itu, kita menghitung persamaan karakteristik dengan menyelesaikan determinan matriks berikut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-30678afffed54546baac35a9eeda7e74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}-1-\\lambda&amp;3&amp;1\\\\[1.1ex] 0&amp;2-\\lambda&amp;0\\\\[1.1ex] 3&amp;-1&amp;1-\\lambda \\end{vmatrix} = -\\lambda^3+2\\lambda^2+4\\lambda-8\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"473\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang kita harus menghitung akar-akar polinomial minimum. Karena merupakan polinomial derajat ketiga, kita menerapkan aturan Ruffini untuk memfaktorkannya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1022b20e607032ce89202906035a1315_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{array}{r|rrrr} &amp; -1&amp;2&amp;\\phantom{-}4&amp; -8 \\\\[2ex] 2 &amp; &amp; -2&amp;0&amp;8 \\\\ \\hline &amp;-1\\vphantom{\\Bigl)}&amp;0&amp;4&amp;0 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"93\" width=\"181\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kemudian kita menemukan akar polinomial yang diperoleh:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef5a87ff07eca3feb9798f85cd0b21c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -\\lambda^2+4=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda = +2 \\\\[2ex] \\lambda = -2 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi nilai eigen matriksnya adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-76aa799dc37e1ba9c8839ac219e2047f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda=2 \\qquad \\lambda =2 \\qquad \\lambda = -2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"213\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nilai eigen dari -2 adalah multiplisitas aljabar sederhana, sebaliknya nilai eigen dari 2 adalah multiplisitas ganda.<\/p>\n<p class=\"has-text-align-left\"> Setelah nilai eigen ditemukan, kami menghitung vektor eigen yang terkait dengan masing-masing nilai tersebut. Pertama, vektor eigen yang sesuai dengan nilai eigen -2: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-70c2775e4e4ba721178bb0bb01743b0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A+2I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f548e75ebc3648368d043737d26c3141_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}1&amp;3&amp;1\\\\[1.1ex] 0&amp;4&amp;0\\\\[1.1ex] 3&amp;-1&amp;3\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0 \\\\[1.1ex] 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"202\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eda9945255b333b217a9c40fc90fb632_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} x+3y+z = 0 \\\\[2ex] 4y = 0\\\\[2ex] 3x-y+3z = 0\\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l}y=0 \\\\[2ex] x = -z \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"255\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e79ea01eaeac74b4cf803f470fbb329b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] 0 \\\\[1.1ex] -1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang mari kita hitung vektor eigen yang terkait dengan nilai eigen 2. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c6944f71d79a33d4789affbc82db4c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-2I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-53c61e86f8559cae71cca6a111379645_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}-3&amp;3&amp;1\\\\[1.1ex] 0&amp;0&amp;0\\\\[1.1ex] 3&amp;-1&amp;-1\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0 \\\\[1.1ex] 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"229\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9de45793fef0fd80dff4c8013e9d444d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -3x+3y+z = 0 \\\\[2ex] 0= 0\\\\[2ex] 3x-y-z = 0\\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l}y=0 \\\\[2ex] z=3x \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"264\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e90d075bde6188e524147bdd92aa203d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] 0 \\\\[1.1ex] 3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Karena nilai eigen 2 diulang dua kali, kita perlu menghitung vektor eigen lain yang memenuhi persamaan subruang (atau ruang eigen):<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-517e77ee4e68f74541ce05ff82fe8188_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}-1 \\\\[1.1ex] 0 \\\\[1.1ex] -3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami membangun matriks<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> , dibentuk oleh tiga vektor eigen matriks:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6a457d6a8a6af3a42596803162118e90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  P = \\begin{pmatrix}1&amp;1&amp;-1 \\\\[1.1ex] 0&amp;0&amp;0 \\\\[1.1ex] -1&amp;3&amp;-3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"150\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Namun ketiga vektor tersebut tidak bebas linier, karena jelas kedua vektor eigen dengan nilai eigen 2 merupakan kombinasi linier satu sama lain. Hal ini juga dapat ditunjukkan karena determinan dari matriks<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> sama dengan 0 (memiliki garis penuh nol):<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eef0b8cbbbfc27e1f11bee978f009064_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(P) = \\begin{vmatrix}1&amp;1&amp;-1 \\\\[1.1ex] 0&amp;0&amp;0 \\\\[1.1ex] -1&amp;3&amp;-3 \\end{vmatrix}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"207\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, karena vektor eigen bergantung linier, <strong>matriks A tidak dapat didiagonalisasi<\/strong> .<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Latihan 5<\/h3>\n<p> Jika memungkinkan, diagonalkan matriks persegi berikut berukuran 3\u00d73: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c00122db1c4520c4ff5907ba29c05647_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}3&amp;0&amp;0\\\\[1.1ex] 0&amp;2&amp;1\\\\[1.1ex] 0&amp;1&amp;2\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"122\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Langkah pertama adalah mencari nilai eigen matriks A. Oleh karena itu, kita menghitung persamaan karakteristik dengan menyelesaikan determinan matriks berikut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0bc2d83752a7ea5c3532047677b123b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}3-\\lambda&amp;0&amp;0\\\\[1.1ex] 0&amp;2-\\lambda&amp;1\\\\[1.1ex] 0&amp;1&amp;2-\\lambda \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"281\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Karena baris pertama seluruhnya terdiri dari nol kecuali 3, kita akan memanfaatkan ini untuk menyelesaikan determinan matriks dengan kofaktor (atau adjoin):<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f27c9abc6047b2289c6dca75524c36b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix}3-\\lambda&amp;0&amp;0\\\\[1.1ex] 0&amp;2-\\lambda&amp;1\\\\[1.1ex] 0&amp;1&amp;2-\\lambda \\end{vmatrix}&amp; = (3-\\lambda)\\cdot  \\begin{vmatrix} 2-\\lambda&amp;1\\\\[1.1ex]1&amp;2-\\lambda \\end{vmatrix} \\\\[3ex] &amp; = (3-\\lambda)[\\lambda^2 -4\\lambda +3] \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"136\" width=\"364\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang kita perlu menghitung akar-akar polinomial karakteristik. Sebaiknya jangan mengalikan tanda kurung karena Anda akan mendapatkan polinomial derajat ketiga. Sebaliknya, jika kedua faktor diselesaikan secara terpisah, maka akan lebih mudah untuk mendapatkan nilai eigennya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-51bd286d6b714a75da7b952b21b01000_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (3-\\lambda)[\\lambda^2 -4\\lambda +3]=0 \\ \\longrightarrow \\ \\begin{cases} 3-\\lambda=0 \\ \\longrightarrow \\ \\lambda = 3 \\\\[2ex] \\lambda^2 -4\\lambda +3=0 \\ \\longrightarrow \\begin{cases}\\lambda = 1 \\\\[2ex] \\lambda = 3 \\end{cases} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"476\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi nilai eigen matriksnya adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0cab1e45f633f7419506c6af08ec1f6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda=1 \\qquad \\lambda =3 \\qquad \\lambda = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"200\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Setelah nilai eigen ditemukan, kami menghitung vektor eigen yang terkait dengan masing-masing nilai tersebut. Pertama, vektor eigen yang sesuai dengan nilai eigen 1: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-10506efea4c355e8449378bc3a1948a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e830c9d5e670fac1f34cbd469a11255_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}2&amp;0&amp;0\\\\[1.1ex] 0&amp;1&amp;1\\\\[1.1ex] 0&amp;1&amp;1\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0 \\\\[1.1ex] 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"188\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a2afb8e9c13b45197cd1b96c25dd7f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 2x = 0 \\\\[2ex] y+z = 0\\\\[2ex] y+z = 0\\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l}x=0 \\\\[2ex] y = -z \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"205\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e82a93f938d6438a3f8caf32715cc3d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}0 \\\\[1.1ex] -1 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kemudian kita menghitung vektor eigen yang terkait dengan nilai eigen 3: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-50e802072a0f6e2942bc873d6a466909_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-3I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4e32cf06d1621a90bae143448d4fa348_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}0&amp;0&amp;0\\\\[1.1ex] 0&amp;-1&amp;1\\\\[1.1ex] 0&amp;1&amp;-1\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0 \\\\[1.1ex] 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d52817f384fe99c1ecc4dce8034d138f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 0 = 0 \\\\[2ex] -y+z = 0\\\\[2ex] y-z = 0\\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l}y=z  \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"205\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f56725fbe621b829ccd3de6e289af91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}0 \\\\[1.1ex] 1 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Karena nilai eigen 3 diulang dua kali, kita perlu menghitung vektor eigen lain yang memenuhi persamaan ruang eigen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d53a91ff3ef0a02d62956e7517bff871_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] 0 \\\\[1.1ex] 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami membangun matriks<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> , dibentuk oleh vektor eigen matriks:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-01c9824b06a22d012e8d7f7d10b3d411_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  P = \\begin{pmatrix}0&amp;0&amp;1 \\\\[1.1ex] -1&amp;1&amp;0 \\\\[1.1ex] 1&amp;1&amp;0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"137\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Berbeda dengan latihan 4, dalam kasus ini kita mampu membentuk 3 vektor bebas linier meskipun multiplisitas aljabar dari nilai eigen 3 adalah dua kali lipat. Hal ini dapat dibuktikan dengan melihat determinan matriks tersebut<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> memberikan hasil yang berbeda dari 0:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5b64f45987701e73cd19b7ca0183e20f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(P) = \\begin{vmatrix}0&amp;0&amp;1 \\\\[1.1ex] -1&amp;1&amp;0 \\\\[1.1ex] 1&amp;1&amp;0 \\end{vmatrix} =-2 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"239\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dengan demikian kita dapat melakukan dekomposisi diagonal dari matriks A. Dan matriks diagonal yang bersesuaian adalah matriks yang memiliki nilai eigen pada diagonal utama:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-428f628ac9ba4c7ae6eb615b0e726735_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle D= \\begin{pmatrix}1&amp;0&amp;0\\\\[1.1ex] 0&amp;3&amp;0 \\\\[1.1ex] 0&amp;0&amp;3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"124\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ingatlah bahwa nilai eigen harus ditempatkan dalam urutan yang sama dengan penempatan vektor eigen dalam matriks<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> .<\/p>\n<p class=\"has-text-align-left\"> Singkatnya, matriks perubahan basis yang diperlukan untuk mendiagonalisasi matriks dan bentuk diagonalnya adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d6fc326a197ddeb33da66d0ecbb5f3b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle P = \\begin{pmatrix}0&amp;0&amp;1 \\\\[1.1ex] -1&amp;1&amp;0 \\\\[1.1ex] 1&amp;1&amp;0 \\end{pmatrix}\\qquad D= \\begin{pmatrix}1&amp;0&amp;0\\\\[1.1ex] 0&amp;3&amp;0 \\\\[1.1ex] 0&amp;0&amp;3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"304\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 6<\/h3>\n<p> Jika memungkinkan, lakukan diagonalisasi matriks berdimensi 4\u00d74 berikut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b25cef0f514564a0206c2f8a588bd346_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix}2&amp;1&amp;2&amp;0\\\\[1.1ex] 1&amp;-3&amp;1&amp;0\\\\[1.1ex] 0&amp;-1&amp;0&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;5\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"161\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Langkah pertama adalah mencari nilai eigen matriks A. Oleh karena itu, kita menghitung persamaan karakteristik dengan menyelesaikan determinan matriks berikut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7cacf7f3b5f63ab816368aaa866e5762_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}2-\\lambda&amp;1&amp;2&amp;0\\\\[1.1ex] 1&amp;-3-\\lambda&amp;1&amp;0\\\\[1.1ex] 0&amp;-1&amp;-\\lambda&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;5-\\lambda\\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"335\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dalam hal ini, kolom terakhir determinan hanya terdiri dari nol kecuali satu elemen, oleh karena itu kita akan memanfaatkan ini untuk menghitung determinan berdasarkan kofaktor melalui kolom ini:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4211dd57b179125aa12310419b051ccb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix}2-\\lambda&amp;1&amp;2&amp;0\\\\[1.1ex] 1&amp;-3-\\lambda&amp;1&amp;0\\\\[1.1ex] 0&amp;-1&amp;-\\lambda&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;5-\\lambda\\end{vmatrix}&amp; = (5-\\lambda)\\cdot  \\begin{vmatrix}2-\\lambda&amp;1&amp;2\\\\[1.1ex] 1&amp;-3-\\lambda&amp;1\\\\[1.1ex] 0&amp;-1&amp;-\\lambda\\end{vmatrix}\\\\[3ex] &amp; = (5-\\lambda)[-\\lambda^3 -\\lambda^2 +6\\lambda] \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"161\" width=\"472\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang kita harus menghitung akar-akar polinomial karakteristik. Sebaiknya jangan mengerjakan perkalian tanda kurung karena Anda akan mendapatkan polinomial derajat keempat. Namun, jika kedua faktor tersebut diselesaikan secara terpisah, akan lebih mudah untuk menghitung nilai eigennya: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c85f64406d449b4f23e6bbc31ee093b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (5-\\lambda)[-\\lambda^3 -\\lambda^2 +6\\lambda]=0 \\ \\longrightarrow \\ \\begin{cases} 5-\\lambda=0 \\ \\longrightarrow \\ \\lambda = 5 \\\\[2ex] -\\lambda^3 -\\lambda^2 +6\\lambda =0 \\ \\longrightarrow \\ \\lambda(-\\lambda^2 -\\lambda +6) =0 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"593\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-96a7a939f5e7d075a94581b2354f7c79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda(-\\lambda^2 -\\lambda +6)=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda=0  \\\\[2ex] -\\lambda^2 -\\lambda +6=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda=2 \\\\[2ex] \\lambda = -3 \\end{cases}\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"467\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi nilai eigen matriksnya adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a035c19d3bf8a877933101ccb35189c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda=0 \\qquad \\lambda =-3 \\qquad \\lambda = 2\\qquad \\lambda = 5\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"291\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Setelah semua nilai eigen ditemukan, kita beralih ke vektor eigen. Kami menghitung vektor eigen yang terkait dengan nilai eigen 0: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0e3b04137690f84b723e3ed568e1114a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-0I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-48e7a88c722f93154455a7d3a139e9e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 2&amp;1&amp;2&amp;0\\\\[1.1ex] 1&amp;-3&amp;1&amp;0\\\\[1.1ex] 0&amp;-1&amp;0&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;5\\end{pmatrix}\\begin{pmatrix}w \\\\[1.1ex] x \\\\[1.1ex] y\\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0\\\\[1.1ex] 0 \\\\[1.1ex] 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"230\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bd06b2978a9da318f23d71c96d5d028e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 2w+x+2y = 0 \\\\[2ex] w-3x+y = 0\\\\[2ex] -x=0 \\\\[2ex] 5z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} x=0 \\\\[2ex] z=0  \\\\[2ex]w=-y \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"262\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5e6cb2192b1819fcd5216e1ad0b37346_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}-1 \\\\[1.1ex] 0 \\\\[1.1ex] 1  \\\\[1.1ex]0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen -3: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a30172d2befd05d52d80c2792c8b917f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A+3I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bb4e1e57896d33ad465b139fee1f0069_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 5&amp;1&amp;2&amp;0\\\\[1.1ex] 1&amp;0&amp;1&amp;0\\\\[1.1ex] 0&amp;-1&amp;3&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;8\\end{pmatrix}\\begin{pmatrix}w \\\\[1.1ex] x \\\\[1.1ex] y\\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0\\\\[1.1ex] 0 \\\\[1.1ex] 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"230\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0190aad5028c9efdaebb2226b863104d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 5w+x+2y = 0 \\\\[2ex] w+y = 0\\\\[2ex] -x+3y=0 \\\\[2ex] 8z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} w=-y  \\\\[2ex]x=3y \\\\[2ex] z=0 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"262\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1d4c0c3b06a7cdb14076a2d1dc0eb395_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}-1 \\\\[1.1ex] 3 \\\\[1.1ex] 1  \\\\[1.1ex]0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen 2: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c6944f71d79a33d4789affbc82db4c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-2I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7263a76e7855eedadeecb32ac4e3a097_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 0&amp;1&amp;2&amp;0\\\\[1.1ex] 1&amp;-5&amp;1&amp;0\\\\[1.1ex] 0&amp;-1&amp;-2&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;3\\end{pmatrix}\\begin{pmatrix}w \\\\[1.1ex] x \\\\[1.1ex] y\\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0\\\\[1.1ex] 0 \\\\[1.1ex] 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"244\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b1655d285c99ec5c316ac5b56f7a2bfb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} x+2y = 0 \\\\[2ex] w-5x+y = 0\\\\[2ex] -x-2y=0 \\\\[2ex] 3z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} x=-2y \\\\[2ex] w=-11y \\\\[2ex] z=0  \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"271\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b72552cd6d30c1f940b2c8ebefa911f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}-11 \\\\[1.1ex] -2 \\\\[1.1ex] 1  \\\\[1.1ex]0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"91\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen 5: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f48052a078660236820e9f605996e193_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-5I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cad6a424d357b8ab8ad0dbf5b6a9a1fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} -3&amp;1&amp;2&amp;0\\\\[1.1ex] 1&amp;-8&amp;1&amp;0\\\\[1.1ex] 0&amp;-1&amp;-5&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;0\\end{pmatrix}\\begin{pmatrix}w \\\\[1.1ex] x \\\\[1.1ex] y\\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0\\\\[1.1ex] 0 \\\\[1.1ex] 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"258\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3531f26937c3668fb457e0af0cf8761d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -3w+x+2y = 0 \\\\[2ex] w-8x+y = 0\\\\[2ex] -x-5y=0 \\\\[2ex] 0=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} w=x=y=0 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"329\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-633b2852390bdc22c60e2aaf38b6ab2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}0 \\\\[1.1ex] 0 \\\\[1.1ex] 0 \\\\[1.1ex]1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami membuat matriks<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> , terdiri dari vektor eigen matriks:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d01ab11cb87f40e42c259bf37e95130f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  P = \\begin{pmatrix}-1&amp;-1&amp;-11&amp;0 \\\\[1.1ex] 0&amp;3&amp;-2&amp;0 \\\\[1.1ex] 1&amp;1&amp;1&amp;0  \\\\[1.1ex]0&amp;0&amp;0&amp;1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"198\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Karena semua nilai eigen berbeda satu sama lain, matriks A dapat didiagonalisasi. Jadi matriks diagonal yang bersesuaian adalah matriks yang mempunyai nilai eigen pada diagonal utama:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8174826f72dc49491c2884f32f54febf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle D= \\begin{pmatrix}0&amp;0&amp;0&amp;0\\\\[1.1ex] 0&amp;-3&amp;0&amp;0 \\\\[1.1ex] 0&amp;0&amp;2&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;5\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"163\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ingatlah bahwa nilai eigen harus ditempatkan dalam urutan yang sama dengan penempatan vektor eigen dalam matriks<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> .<\/p>\n<p class=\"has-text-align-left\"> Ringkasnya, perubahan matriks dasar yang diperlukan untuk mendiagonalisasi matriks A dan matriks dalam bentuk diagonal adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-029be0f37d9f5846758b7dbb1e25c8fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle P = \\begin{pmatrix}-1&amp;-1&amp;-11&amp;0 \\\\[1.1ex] 0&amp;3&amp;-2&amp;0 \\\\[1.1ex] 1&amp;1&amp;1&amp;0  \\\\[1.1ex]0&amp;0&amp;0&amp;1\\end{pmatrix} \\qquad D=\\begin{pmatrix}0&amp;0&amp;0&amp;0\\\\[1.1ex] 0&amp;-3&amp;0&amp;0 \\\\[1.1ex] 0&amp;0&amp;2&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;5\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"404\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-118\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"> Penerapan matriks yang dapat didiagonalisasi<\/h2>\n<p> Jika Anda sudah sampai sejauh ini, Anda mungkin bertanya-tanya: untuk apa matriks yang dapat didiagonalisasi?<\/p>\n<p class=\"has-text-align-left\"> Nah, matriks yang dapat didiagonalisasi sangat berguna dan banyak digunakan dalam matematika. Alasannya adalah matriks diagonal praktis penuh dengan angka nol sehingga membuat perhitungan menjadi lebih mudah.<\/p>\n<p> Contoh nyata dari hal ini adalah <strong>pangkat matriks yang dapat didiagonalisasi,<\/strong> karena hasilnya disederhanakan dengan rumus berikut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a58b001c11304f21fbb6c1f2ac53766f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^k=PD^kP^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"113\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Kesetaraan ini dapat dengan mudah dibuktikan dengan induksi. Oleh karena itu cukup untuk menaikkan matriks<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b9ef1bbd23fd1b198de883813285620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> kepada peserta pameran. Dan karena ini adalah matriks diagonal, operasinya dikurangi menjadi menaikkan setiap suku diagonal utama menjadi eksponen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-61f8a7778e43eedecad71920e45f7471_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  D^k = diag(\\lambda_1^k,\\lambda_2^k, \\ldots , \\lambda_n^k)\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"198\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Contoh pangkat matriks yang dapat didiagonalisasi<\/h3>\n<p> Untuk lebih memahaminya, kita akan menghitung pangkat matriks yang dapat didiagonalisasi sebagai contoh:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3544a0199a7c277c7497a042deee07ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}2&amp;0\\\\[1.1ex] 3&amp;1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"95\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Matriks Perubahan Dasar<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> , dibentuk oleh vektor eigennya, dan matriks yang didiagonalisasi<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b9ef1bbd23fd1b198de883813285620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> , terdiri dari nilai-nilainya sendiri, adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d5f3ac30ba6b6ac40e819e86daad73e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle P = \\begin{pmatrix}0&amp;1 \\\\[1.1ex] 1&amp;3 \\end{pmatrix} \\qquad D= \\begin{pmatrix}1&amp;0\\\\[1.1ex] 0&amp;2\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"235\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Jadi, sebagai contoh, matriks A yang dipangkatkan menjadi 7 ekuivalen dengan:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-474e92843d1a973a45a0cfe8fc8889ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^7=PD^7P^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"112\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c0cbd00c0e1c04f294d8ff5413894e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^7=\\begin{pmatrix}0&amp;1 \\\\[1.1ex] 1&amp;3\\end{pmatrix}\\begin{pmatrix}1&amp;0\\\\[1.1ex] 0&amp;2\\end{pmatrix}^7\\left.\\begin{pmatrix}0&amp;1 \\\\[1.1ex] 1&amp;3 \\end{pmatrix}\\right.^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"265\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Sekarang kita membalikkan matriksnya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7868fd8a15a99bfc9b31b1e4732bcc8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9747b331e1548549fa7a171695729eec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^7=\\begin{pmatrix}0&amp;1 \\\\[1.1ex] 1&amp;3 \\end{pmatrix}\\begin{pmatrix}1&amp;0\\\\[1.1ex] 0&amp;2\\end{pmatrix}^7\\begin{pmatrix}-3&amp;1 \\\\[1.1ex] 1&amp;0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"254\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Kami memecahkan kekuatan matriks <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0678df2cc9faf293040c255b8d05014d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-147918af8d66f941dcd70444b7e0d5a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^7=\\begin{pmatrix}0&amp;1 \\\\[1.1ex] 1&amp;3\\end{pmatrix}\\begin{pmatrix}1^7&amp;0\\\\[1.1ex] 0&amp;2^7\\end{pmatrix} \\begin{pmatrix}-3&amp;1 \\\\[1.1ex] 1&amp;0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"262\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-27c7f9dee1b20761a9845457099573cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^7=\\begin{pmatrix}0&amp;1 \\\\[1.1ex] 1&amp;3 \\end{pmatrix}\\begin{pmatrix}1&amp;0\\\\[1.1ex] 0&amp;128\\end{pmatrix} \\begin{pmatrix}-3&amp;1 \\\\[1.1ex] 1&amp;0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"264\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dan terakhir, kita melakukan perkalian matriks:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f77fa7a83f343c5723afa0a3fde981cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{A^7=}\\begin{pmatrix}\\bm{128}&amp;\\bm{0}\\\\[1.1ex] \\bm{381}&amp;\\bm{1}\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"118\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Seperti yang Anda lihat, menghitung pangkat dengan matriks diagonal lebih mudah daripada mengalikan matriks yang sama tujuh kali berturut-turut. Lalu bayangkan dengan nilai eksponen yang jauh lebih besar. <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-119\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"> Sifat-sifat matriks yang dapat didiagonalisasi<\/h2>\n<p> Ciri-ciri matriks jenis ini adalah:<\/p>\n<ul>\n<li> Jika matriks\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> dapat didiagonalisasi, kekuatan apa pun<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> .<\/li>\n<\/ul>\n<ul>\n<li> Hampir semua matriks dapat didiagonalisasi pada lingkungan yang kompleks\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-68da13602f004ced593a0442bca3f363_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mathbb{C}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> . Meskipun di bawah ini Anda memiliki pengecualian yang tidak pernah dapat didiagonalisasi.<\/li>\n<\/ul>\n<ul>\n<li> Jika matriks\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> adalah matriks ortogonal, maka kita katakan matriks tersebut<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> <strong>dapat didiagonalisasi secara ortogonal<\/strong> dan, oleh karena itu, persamaannya dapat ditulis ulang:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e3f65f9edb18ea2a563767416aec8e52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A=PDP^t\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Suatu matriks dapat didiagonalisasi oleh matriks kesatuan jika dan hanya jika matriks tersebut merupakan matriks normal.<\/li>\n<\/ul>\n<ul>\n<li> Diberikan dua matriks yang dapat didiagonalisasi, matriks-matriks tersebut dapat diubah jika dan hanya jika matriks-matriks tersebut dapat didiagonalisasi secara bersamaan, yaitu jika matriks-matriks tersebut memiliki basis vektor eigen (atau vektor eigen) ortonormal yang sama.<\/li>\n<\/ul>\n<ul>\n<li> Jika suatu endomorfisme dapat didiagonalisasi, kita katakan bahwa endomorfisme tersebut <strong>dapat didiagonalisasi karena kesamaan<\/strong> . Namun, tidak semua endomorfisme dapat didiagonalisasi, atau dengan kata lain, diagonalisasi suatu endomorfisme tidak terjamin.<\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"> Diagonalisasi Serentak<\/h2>\n<p> Suatu himpunan matriks dikatakan <strong>dapat didiagonalisasi secara serentak<\/strong> jika terdapat matriks yang dapat dibalik yang menjadi dasar untuk mendiagonalisasi matriks mana pun dalam himpunan tersebut. Dengan kata lain, jika dua matriks didiagonalisasi berdasarkan vektor eigen yang sama, berarti matriks-matriks tersebut dapat didiagonalisasi secara bersamaan.<\/p>\n<p> Selain itu, seperti yang telah kami komentari dalam sifat-sifat diagonalisasi matriks, jika dua matriks mampu melakukan diagonalisasi secara bersamaan, keduanya harus saling berpindah-pindah.<\/p>\n<p> Misalnya, dua matriks berikut dapat diubah, sehingga keduanya didiagonalisasi berdasarkan vektor eigen atau vektor eigen yang sama.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c215d8b5d9ae75dbd069c6b6d39886dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A=\\begin{pmatrix}2&amp;0 \\\\[1.1ex] 1&amp;-1 \\end{pmatrix} \\qquad B=\\begin{pmatrix}3&amp;0\\\\[1.1ex] 1&amp;0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"247\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Sekalipun keduanya memiliki vektor eigen yang sama, bukan berarti keduanya mempunyai nilai eigen yang sama. Faktanya, meskipun matriks A dan B di atas memiliki vektor eigen yang sama, namun nilai eigennya berbeda.<\/p>\n<h2 class=\"wp-block-heading\"> Matriks yang tidak dapat didiagonalisasi<\/h2>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<p> Meskipun sebagian besar matriks dapat didiagonalisasi dalam lingkungan bilangan kompleks, <strong>beberapa matriks tidak pernah dapat didiagonalisasi.<\/strong><\/p>\n<p> Fakta ini terjadi ketika multiplisitas aljabar suatu nilai eigen (atau nilai eigen) tidak sesuai dengan multiplisitas geometri.<\/p>\n<p> Misalnya, matriks berikut tidak dapat didiagonalisasi dengan cara apa pun, melainkan \u201cdapat didiagonalisasi\u201d:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cdadeeadd8ee984e2efb53896c2d3306_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}0&amp;1 \\\\[1.1ex] 0&amp;0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"54\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Selain itu, ada matriks yang tidak mampu melakukan diagonalisasi dalam lingkungan bilangan real, tetapi matriks tersebut melakukan diagonalisasi ketika bekerja dengan bilangan kompleks, seperti matriks ini:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c7bb0fa6573d760edc55d94cfc834c7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{pmatrix}0&amp;1 \\\\[1.1ex] -1&amp;0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Terakhir, ada beberapa prosedur <em>diagonalisasi blok matriks<\/em> yang tidak sepenuhnya dapat didiagonalisasi, namun sedikit lebih rumit. Metode yang paling terkenal adalah diagonalisasi dengan <a href=\"https:\/\/es.wikipedia.org\/wiki\/Forma_can%C3%B3nica_de_Jordan\" target=\"_blank\" rel=\"noreferrer noopener\">bentuk kanonik Jordan<\/a> .<\/p>\n<h2 class=\"wp-block-heading\"> Mendiagonalisasi matriks dengan MATLAB<\/h2>\n<p> Program komputer sangat berguna dalam mendiagonalisasi matriks, terutama jika matriksnya sangat besar. Dan software yang paling terkenal tentunya <strong>MATLAB<\/strong> , jadi selanjutnya kita akan melihat cara memfaktorkan matriks secara diagonal menggunakan program ini.<\/p>\n<p> Instruksi yang digunakan untuk mendiagonalisasi matriks dengan MATLAB adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d9c2b022364c0099b96b150c5853a9f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\text{[P, D] = eig(A)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Emas<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> adalah matriks yang akan didiagonalisasi dan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b9ef1bbd23fd1b198de883813285620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> adalah matriks yang dikembalikan oleh program:<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> adalah matriks yang dibentuk oleh vektor eigen dan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b9ef1bbd23fd1b198de883813285620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> adalah matriks berbentuk diagonal yang suku diagonal utamanya adalah nilai eigennya.<\/p>\n<p> Oleh karena itu, Anda hanya perlu memasukkan kode ini ke dalam program.<\/p>\n<p> Sebaliknya, jika Anda hanya ingin mengetahui nilai eigen saja, Anda dapat menggunakan pernyataan berikut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1e908bfbcd51e3b8c338b5ca279f9f8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  e= eig(A)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Emas<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3fc193f43cc29c1eef788f64ba43c1bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"e\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> adalah vektor kolom yang dikembalikan MATLAB dengan nilai eigen matriks<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> .<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Di halaman ini Anda akan menemukan segala sesuatu tentang matriks yang dapat didiagonalisasi: apa itu matriks, kapan matriks tersebut dapat didiagonalisasi dan kapan tidak dapat didiagonalisasi, metode untuk mendiagonalisasi matriks, penerapan dan sifat-sifat matriks tertentu, dll. Dan Anda bahkan memiliki beberapa latihan yang diselesaikan selangkah demi selangkah sehingga Anda dapat berlatih dan memahami dengan sempurna &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/cara-mendiagonalisasi-matriks-yang-dapat-didiagonalisasi-latihan-diagonalisasi-matriks-2x2-3x3-4x4-diselesaikan-langkah-demi-langkah\/\"> <span class=\"screen-reader-text\">Cara mendiagonalisasi matriks<\/span> Selengkapnya &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[39],"tags":[],"class_list":["post-68","post","type-post","status-publish","format-standard","hentry","category-penentu-suatu-matriks"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Cara mendiagonalisasi matriks -<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/id\/cara-mendiagonalisasi-matriks-yang-dapat-didiagonalisasi-latihan-diagonalisasi-matriks-2x2-3x3-4x4-diselesaikan-langkah-demi-langkah\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Cara mendiagonalisasi matriks -\" \/>\n<meta property=\"og:description\" content=\"Di halaman ini Anda akan menemukan segala sesuatu tentang matriks yang dapat didiagonalisasi: apa itu matriks, kapan matriks tersebut dapat didiagonalisasi dan kapan tidak dapat didiagonalisasi, metode untuk mendiagonalisasi matriks, penerapan dan sifat-sifat matriks tertentu, dll. 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