{"id":344,"date":"2023-07-06T02:49:35","date_gmt":"2023-07-06T02:49:35","guid":{"rendered":"https:\/\/mathority.org\/id\/contoh-matriks-simetris-dan-sifat-sifatnya\/"},"modified":"2023-07-06T02:49:35","modified_gmt":"2023-07-06T02:49:35","slug":"contoh-matriks-simetris-dan-sifat-sifatnya","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/contoh-matriks-simetris-dan-sifat-sifatnya\/","title":{"rendered":"Matriks simetris"},"content":{"rendered":"<p>Di halaman ini Anda akan menemukan penjelasan tentang apa itu matriks simetris. Selain itu, kami akan menunjukkan kepada Anda cara cepat mengidentifikasi suatu matriks simetris, beserta beberapa contohnya sehingga Anda tidak perlu ragu. Anda juga akan menemukan semua sifat matriks simetris. Dan terakhir, kami menjelaskan ciri khusus yang dimiliki oleh setiap matriks persegi: matriks tersebut dapat diuraikan menjadi jumlah matriks simetris dan matriks antisimetris.<\/p>\n<h2 class=\"wp-block-heading\"> Apa itu matriks simetris?<\/h2>\n<p> Pengertian matriks simetris adalah sebagai berikut: <\/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 simetris<\/strong> adalah matriks persegi yang transposnya sama dengan matriks itu sendiri.<\/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-9d91251629a6f0241682eed5c4d82847_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^t = A\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" 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-afd3cedfe0f405ed9f2d585b5ac1d8cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^t\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<p> mewakili matriks yang ditransposisikan 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> Setelah kita mengetahui konsep matriks simetris, kita akan melihat bagaimana matriks simetris dapat dengan mudah diidentifikasi:<\/p>\n<h2 class=\"wp-block-heading\"> Kapan suatu matriks simetris?<\/h2>\n<p> Mengenali struktur matriks simetris sangatlah sederhana: elemen baris <em>i<\/em> dan kolom <em>j<\/em> harus identik dengan elemen baris <em>j<\/em> dan kolom <em>i<\/em> . Dan nilai diagonal utama matriks dapat berupa apa saja.<\/p>\n<h2 class=\"wp-block-heading\"> Contoh matriks simetris<\/h2>\n<p> Berikut beberapa contoh matriks simetris untuk membantu Anda memahami:<\/p>\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#1976d2\"> <span style=\"text-decoration: underline;\">Contoh matriks simetris berorde 2\u00d72<\/span> <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-de-matrice-symetrique-de-dimension-22152-1.webp\" alt=\"contoh matriks simetris berdimensi 2x2\" class=\"wp-image-3524\" width=\"77\" height=\"71\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#1976d2\"> <span style=\"text-decoration: underline;\">Contoh matriks simetris berdimensi 3\u00d73<\/span> <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-de-matrice-tridimensionnelle-symetrique3-1.webp\" alt=\"contoh matriks simetris berdimensi 3x3\" class=\"wp-image-3525\" width=\"112\" height=\"118\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#1976d2\"> <span style=\"text-decoration: underline;\">Contoh matriks simetris berukuran 4\u00d74<\/span> <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-de-matrice-symetrique-de-dimension-42154-1.webp\" alt=\"contoh matriks simetris berdimensi 4x4\" class=\"wp-image-3526\" width=\"206\" height=\"138\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Dengan melakukan transposisi ketiga matriks ini kita memverifikasi bahwa matriks-matriks tersebut simetris, karena matriks-matriks yang ditransposisikan ekuivalen dengan matriks aslinya masing-masing.<\/p>\n<h2 class=\"wp-block-heading\"> Mengapa disebut matriks simetris?<\/h2>\n<p> Jika diperhatikan contoh sebelumnya, diagonal utama suatu matriks simetris adalah sumbu simetri, atau dengan kata lain bertindak sebagai cermin antara bilangan di atas diagonal dan bilangan di bawahnya. Oleh karena itu, matriks jenis ini disebut simetris.<\/p>\n<h2 class=\"wp-block-heading\"> Sifat-sifat matriks simetris<\/h2>\n<p> Ciri-ciri matriks simetris adalah sebagai berikut:<\/p>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<ul>\n<li> Penjumlahan (atau pengurangan) dua matriks simetris menghasilkan matriks simetris lainnya. Karena mentransposisi dua matriks yang ditambah (atau dikurangi) sama dengan mentransposisi setiap matriks secara terpisah:<\/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-9942bb6c2d0b3b406e42f6b1365e7151_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(A+B\\right)^t = A^t+B^t = A+B\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"225\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Setiap matriks simetris yang dikalikan dengan skalar juga akan menghasilkan matriks simetris lainnya.<\/li>\n<\/ul>\n<ul>\n<li> Demikian pula hasil kali matriks antara dua matriks simetris tidak selalu sama dengan matriks simetris lainnya, hanya jika dan hanya jika kedua matriks tersebut dapat dikomutasi. Kondisi ini dapat dibuktikan dengan sifat perkalian matriks yang ditransposisikan:<\/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-03f81c2643b3093a4db891724660c3b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(A\\cdot B\\right)^t = B^t\\cdot A^t = BA=AB\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"237\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Pangkat suatu matriks simetris memunculkan matriks simetris lainnya, asalkan eksponennya bilangan bulat.<\/li>\n<\/ul>\n<ul>\n<li> Jelasnya, <a href=\"https:\/\/mathority.org\/id\">matriks kesatuan<\/a> dan matriks nol merupakan contoh matriks simetris.<\/li>\n<\/ul>\n<ul>\n<li> Suatu matriks yang kongruen dengan matriks simetris juga harus simetris.<\/li>\n<\/ul>\n<ul>\n<li> Jika suatu matriks simetris beraturan atau dapat dibalik, maka matriks inversnya juga simetris.<\/li>\n<\/ul>\n<ul>\n<li> Begitu pula dengan adjoin matriks simetris: matriks adjoin dari matriks simetris menghasilkan matriks simetris lain sebagai solusinya.<\/li>\n<\/ul>\n<ul>\n<li> Matriks simetris sejati juga merupakan matriks normal.<\/li>\n<\/ul>\n<ul>\n<li> Karena matriks simetris merupakan kasus khusus dari matriks Hermitian, semua nilai eigen (atau nilai eigen) dari matriks simetris adalah bilangan real.<\/li>\n<\/ul>\n<ul>\n<li> Teorema spektral menyatakan bahwa semua matriks yang elemen-elemennya nyata adalah matriks yang dapat didiagonalisasi dan, terlebih lagi, diagonalisasi dilakukan melalui matriks ortogonal. Oleh karena itu, semua matriks simetris nyata didiagonalisasi secara ortogonal.<\/li>\n<\/ul>\n<ul>\n<li> Sebaliknya, matriks simetris dengan bilangan kompleks dapat didiagonalisasi melalui matriks kesatuan.<\/li>\n<\/ul>\n<ul>\n<li> Matriks Hessian selalu simetris. <\/li>\n<\/ul>\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\"> Penguraian matriks persegi menjadi matriks simetris dan matriks antisimetris<\/h2>\n<p> Ciri khusus matriks persegi adalah matriks tersebut dapat diuraikan menjadi jumlah matriks simetris ditambah matriks antisimetris.<\/p>\n<p> Rumus yang memungkinkan kita melakukan hal ini adalah 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-a3b9aa2b7ed0e9ce31587d4f00f1144e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{array}{c} C = S + A \\\\[2ex] S = \\cfrac{1}{2}\\cdot (C+C^t) \\qquad A = \\cfrac{1}{2} \\cdot (C-C^t)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"76\" width=\"293\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dimana C adalah matriks persegi yang ingin kita dekomposisi, C <sup>t<\/sup> transposnya, dan terakhir S dan A masing-masing adalah matriks simetris dan antisimetris yang menjadi tempat dekomposisi matriks C.<\/p>\n<p> Di bawah ini Anda memiliki latihan yang telah diselesaikan untuk melihat bagaimana hal ini dilakukan. Mari kita dekomposisi 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-852a7267895a7f332ad3f28f8a8dda0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle C=\\begin{pmatrix} 2&amp; -1 \\\\[1.1ex] 3 &amp;0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"109\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Kita menghitung matriks simetris dan antisimetris dengan rumus:<\/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-994670ecc17b3bc8757482f1656e543e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle S=\\cfrac{1}{2}\\cdot (C+C^t)= \\begin{pmatrix} 2&amp; 1 \\\\[1.1ex] 1 &amp;0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"210\" 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-d87e1d30d2bc657c20535f45c0fb7be6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\cfrac{1}{2}\\cdot (C-C^t)= \\begin{pmatrix} 0&amp; -2 \\\\[1.1ex] 2 &amp;0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"225\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dan kita dapat memeriksa apakah persamaan tersebut terpenuhi dengan menjumlahkan kedua 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-f2a938eebbcc10adb3c3392634a62fbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle C=S+A \\quad ?\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"110\" style=\"vertical-align: -2px;\"><\/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-7cfdbffec6801c13041cd2996da13e96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{pmatrix} 2&amp; 1 \\\\[1.1ex] 1 &amp;0\\end{pmatrix}+\\begin{pmatrix} 0&amp; -2 \\\\[1.1ex] 2 &amp;0\\end{pmatrix}=\\begin{pmatrix} 2&amp; -1 \\\\[1.1ex] 3 &amp;0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"251\" 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-4d009f71f52fd49559eefc457d18a8be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle C=S+A\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"84\" style=\"vertical-align: -2px;\"><\/p>\n<p> \u2705<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Di halaman ini Anda akan menemukan penjelasan tentang apa itu matriks simetris. Selain itu, kami akan menunjukkan kepada Anda cara cepat mengidentifikasi suatu matriks simetris, beserta beberapa contohnya sehingga Anda tidak perlu ragu. Anda juga akan menemukan semua sifat matriks simetris. Dan terakhir, kami menjelaskan ciri khusus yang dimiliki oleh setiap matriks persegi: matriks tersebut &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/contoh-matriks-simetris-dan-sifat-sifatnya\/\"> <span class=\"screen-reader-text\">Matriks simetris<\/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":[64],"tags":[],"class_list":["post-344","post","type-post","status-publish","format-standard","hentry","category-jenis-tabel"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Matriks simetris - Mathority<\/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\/contoh-matriks-simetris-dan-sifat-sifatnya\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Matriks simetris - Mathority\" \/>\n<meta property=\"og:description\" content=\"Di halaman ini Anda akan menemukan penjelasan tentang apa itu matriks simetris. 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Selain itu, kami akan menunjukkan kepada Anda cara cepat mengidentifikasi suatu matriks simetris, beserta beberapa contohnya sehingga Anda tidak perlu ragu. Anda juga akan menemukan semua sifat matriks simetris. Dan terakhir, kami menjelaskan ciri khusus yang dimiliki oleh setiap matriks persegi: matriks tersebut &hellip; Matriks simetris Selengkapnya &raquo;","og_url":"https:\/\/mathority.org\/id\/contoh-matriks-simetris-dan-sifat-sifatnya\/","article_published_time":"2023-07-06T02:49:35+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9d91251629a6f0241682eed5c4d82847_l3.png"}],"author":"Tim Mathority","twitter_card":"summary_large_image","twitter_misc":{"Ditulis oleh":"Tim Mathority","Estimasi waktu membaca":"3 menit"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/id\/contoh-matriks-simetris-dan-sifat-sifatnya\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/id\/contoh-matriks-simetris-dan-sifat-sifatnya\/"},"author":{"name":"Tim Mathority","@id":"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38"},"headline":"Matriks simetris","datePublished":"2023-07-06T02:49:35+00:00","dateModified":"2023-07-06T02:49:35+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/id\/contoh-matriks-simetris-dan-sifat-sifatnya\/"},"wordCount":553,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/id\/#organization"},"articleSection":["Jenis tabel"],"inLanguage":"id","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/id\/contoh-matriks-simetris-dan-sifat-sifatnya\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/id\/contoh-matriks-simetris-dan-sifat-sifatnya\/","url":"https:\/\/mathority.org\/id\/contoh-matriks-simetris-dan-sifat-sifatnya\/","name":"Matriks simetris - 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