{"id":375,"date":"2023-07-06T06:35:36","date_gmt":"2023-07-06T06:35:36","guid":{"rendered":"https:\/\/mathority.org\/nl\/eigenwaarden-berekenen-eigenwaarden-en-eigenvectoren-eigenvectoren-van-een-matrix\/"},"modified":"2023-07-06T06:35:36","modified_gmt":"2023-07-06T06:35:36","slug":"eigenwaarden-berekenen-eigenwaarden-en-eigenvectoren-eigenvectoren-van-een-matrix","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/eigenwaarden-berekenen-eigenwaarden-en-eigenvectoren-eigenvectoren-van-een-matrix\/","title":{"rendered":"Eigenwaarden (of eigenwaarden) en eigenvectoren (of eigenvectoren) van een matrix"},"content":{"rendered":"<p>Op deze pagina leggen we uit wat eigenwaarden en eigenvectoren zijn, ook wel eigenwaarden en eigenvectoren genoemd. U vindt er ook voorbeelden van hoe u ze kunt berekenen, evenals stapsgewijze opgeloste oefeningen om te oefenen.<\/p>\n<h2 class=\"wp-block-heading\"> Wat is een eigenwaarde en een eigenvector?<\/h2>\n<p> Hoewel het begrip eigenwaarde en eigenvector moeilijk te begrijpen is, is de definitie ervan als volgt: <\/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>Eigenvectoren of eigenvectoren<\/strong> zijn de vectoren die niet nul zijn van een lineaire kaart die, wanneer ze erdoor worden getransformeerd, aanleiding geven tot een scalair veelvoud ervan (ze veranderen niet van richting). Deze scalair is de <strong>eigenwaarde of eigenwaarde<\/strong> .<\/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-710a5e2df8739c35c060f790f5592734_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Av = \\lambda v\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"65\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> Goud<\/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> is de matrix van de lineaire kaart,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"v\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> is de eigenvector en<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2b5c45836864531b8e37025dabadd24a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> eigen waarde.<\/p>\n<\/div>\n<p> Eigenwaarde wordt ook wel karakteristieke waarde genoemd. En er zijn zelfs wiskundigen die de Duitse wortel \u2018eigen\u2019 gebruiken om eigenwaarden en eigenvectoren aan te duiden: <em>eigenwaarden<\/em> voor eigenwaarden en <em>eigenvectoren<\/em> voor eigenvectoren.<\/p>\n<h2 class=\"wp-block-heading\"> Hoe bereken je de eigenwaarden (of eigenwaarden) en de eigenvectoren (of eigenvectoren) van een matrix?<\/h2>\n<p> Om de eigenwaarden en eigenvectoren van een matrix te vinden, moet je een hele procedure volgen:<\/p>\n<ol style=\"color:#1976d2; font-weight: bold;>\n<li><span style=\" color:#262626;font-weight:=\"\" normal;\"=\"\">\n<li style=\"margin-bottom:18px\"><span style=\"color:#262626;font-weight: normal;\">De karakteristieke vergelijking van de matrix wordt berekend door de volgende determinant op te lossen:<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d7224fcfc13d25429e22216a3d4124cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"92\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<li style=\"margin-bottom:15px\"> <span style=\"color:#262626;font-weight: normal;\">We vinden de wortels van de karakteristieke polynoom verkregen in stap 1. Deze wortels zijn de eigenwaarden van de matrix.<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fbe85bd9aff702c72a31d3889f035518_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)=0 \\ \\longrightarrow \\ \\lambda\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"186\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<li style=\"margin-bottom:15px\"> <span style=\"color:#262626;font-weight: normal;\">De eigenvector van elke eigenwaarde wordt berekend. Om dit te doen, wordt voor elke eigenwaarde het volgende stelsel vergelijkingen opgelost:<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-372f1009cb2b47f939cf9291f0f23885_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-\\lambda I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/ol>\n<p> Dit is de methode om de eigenwaarden en eigenvectoren van een matrix te vinden, maar hier geven we je ook enkele tips: \ud83d\ude09 <\/p>\n<div style=\"background-color:#fffde7;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px\" class=\"has-background\">\n<p style=\"text-align:left\"> <strong>Tips<\/strong> : we kunnen profiteren van de eigenschappen van eigenwaarden en eigenvectoren om ze gemakkelijker te berekenen:<\/p>\n<p style=\"text-align:left\"> <strong><span style=\"color:#1976d2;\">\u2713<\/span><\/strong> Het spoor van de matrix (som van de hoofddiagonaal) is gelijk aan de som van alle eigenwaarden.<\/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-d4b8ae7f7f7a36be08403ae6ba8b3d32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle tr(A)=\\sum_{i=1}^n \\lambda_i\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"109\" style=\"vertical-align: -21px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> <strong><span style=\"color:#1976d2;\">\u2713<\/span><\/strong> Het product van alle eigenwaarden is gelijk aan de determinant van de matrix.<\/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-7aa4b68759894e3f25d6475c3b6f71b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle det(A)=\\prod_{i=1}^n \\lambda_i\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"115\" style=\"vertical-align: -21px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> <strong><span style=\"color:#1976d2;\">\u2713<\/span><\/strong> Als er sprake is van een lineaire combinatie tussen rijen of kolommen, is minimaal \u00e9\u00e9n eigenwaarde van de matrix gelijk aan 0.<\/p>\n<\/div>\n<p> Laten we een voorbeeld bekijken van hoe de eigenvectoren en eigenwaarden van een matrix worden berekend om de methode beter te begrijpen:<\/p>\n<h2 class=\"wp-block-heading\"> Voorbeeld van het berekenen van de eigenwaarden en eigenvectoren van een matrix:<\/h2>\n<ul>\n<li> Zoek de eigenwaarden en eigenvectoren van de volgende matrix:<\/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-e82dbe4f6e975e1374cab2c1b74638b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}1&amp;0\\\\[1.1ex] 5&amp;2\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"95\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Eerst moeten we de karakteristieke vergelijking van de matrix vinden. En hiervoor moet de volgende determinant worden opgelost:<\/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-283812fe5eed97f58568fb6e515e3ff5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}1- \\lambda &amp;0\\\\[1.1ex] 5&amp;2-\\lambda \\end{vmatrix} = \\lambda^2-3\\lambda +2\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"338\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Nu berekenen we de wortels van de karakteristieke polynoom, daarom stellen we het verkregen resultaat gelijk aan 0 en lossen we de vergelijking op:<\/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-2287061273d7f8502e0dbf1cb2fe1ad7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda^2-3\\lambda +2 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" 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-fdee5858b8b0187078ea372d9362900f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda= \\cfrac{-(-3)\\pm \\sqrt{(-3)^2-4\\cdot 1 \\cdot 2}}{2\\cdot 1} = \\cfrac{+3\\pm 1}{2}=\\begin{cases} \\lambda = 1 \\\\[2ex] \\lambda = 2 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"419\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> De oplossingen van de vergelijking zijn de eigenwaarden van de matrix.<\/p>\n<p> Zodra we de eigenwaarden hebben, berekenen we de eigenvectoren. Om dit te doen, moeten we voor elke eigenwaarde het volgende systeem oplossen:<\/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-372f1009cb2b47f939cf9291f0f23885_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-\\lambda I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> We berekenen eerst de eigenvector die hoort bij de eigenwaarde 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-372f1009cb2b47f939cf9291f0f23885_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-\\lambda I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" 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-4f0cbd7a7e0670410881dcc0bfd4969c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-1 I)\\begin{pmatrix}x \\\\[1.1ex] y \\end{pmatrix} =}\\begin{pmatrix}0 \\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"163\" 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-e1f49b7ecec643964e4a14cd17ddecb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}0&amp;0\\\\[1.1ex] 5&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=\"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-06473aeaa487551bca2eb98ff786c8f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 0x+0y = 0 \\\\[2ex] 5x+y = 0\\end{array}\\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"112\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Uit deze vergelijkingen verkrijgen we de volgende deelruimte:<\/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-82da1eb92338b5dc67c9e65188b6c247_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle y=-5x\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"66\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Eigenvectordeelruimten worden ook eigenruimten genoemd.<\/p>\n<p> Nu moeten we een basis van deze schone ruimte vinden, dus geven we bijvoorbeeld de waarde 1 aan de variabele<\/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-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> en we verkrijgen de volgende eigenvector: <\/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-f4700ed7bb632b97f0ce1bec12409888_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x = 1 \\ \\longrightarrow \\ y=-5\\cdot 1 = -5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"216\" 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-8af03064a8f197990df832e71472cab0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] -5\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"79\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<p> Ten slotte herhalen we, zodra de eigenvector die bij eigenwaarde 1 hoort, het proces om de eigenvector voor eigenwaarde 2 te berekenen: <\/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-372f1009cb2b47f939cf9291f0f23885_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-\\lambda I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" 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-3d52ccfc2cbc996d3844af6c699a81b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-2I)\\begin{pmatrix}x \\\\[1.1ex] y \\end{pmatrix} =}\\begin{pmatrix}0 \\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"163\" 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-24442a53901cc9f0622aecf66ef2dc25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}-1&amp;0\\\\[1.1ex] 5&amp;0\\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=\"169\" 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-fbd3a434bf3f89ed38a893a98befee97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -x+0y = 0 \\\\[2ex] 5x+0y = 0\\end{array}\\right\\} \\longrightarrow \\ x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"207\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> In dit geval moet alleen de eerste component van de vector 0 zijn, dus we kunnen er elke waarde aan geven<\/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-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> . Maar om het makkelijker te maken is het beter om een 1 te plaatsen:<\/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-f47b6a21a448d003d909c0c1c969b8f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}0 \\\\[1.1ex] 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"66\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Concluderend zijn de eigenwaarden en eigenvectoren van de matrix:<\/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-1668ed5f36ad0a8fcb28a264c76b6163_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 1 \\qquad v = \\begin{pmatrix}1 \\\\[1.1ex] -5 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"158\" 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-56b0287c0bea71a1e5a258373aaa47d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 2 \\qquad v = \\begin{pmatrix}0 \\\\[1.1ex] 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"144\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Als je eenmaal weet hoe je de eigenwaarden en eigenvectoren van een matrix kunt vinden, vraag je je misschien af\u2026 en waar dienen ze voor? Welnu, het blijkt dat ze erg nuttig zijn voor <a href=\"https:\/\/mathority.org\/nl\/hoe-je-een-diagonaliseerbare-matrix-diagonaliseert-diagonalisatie-van-2x2-3x3-4x4-matrixoefeningen-stap-voor-stap-opgelost\/\">matrixdiagonalisatie<\/a> , sterker nog, dat is hun belangrijkste toepassing. Voor meer informatie raden we aan om te kijken hoe je een matrix diagonaliseert met de link, waar de procedure stap voor stap wordt uitgelegd en er ook voorbeelden en opgeloste oefeningen zijn om te oefenen.<\/p>\n<h2 class=\"wp-block-heading\"> Opgeloste oefeningen over eigenwaarden en eigenvectoren (eigenwaarden en eigenvectoren)<\/h2>\n<h3 class=\"wp-block-heading\"> Oefening 1<\/h3>\n<p> Bereken de eigenwaarden en eigenvectoren van de volgende vierkante matrix van orde 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-0c6e3869ea2848140f026afc2ff8d554_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}3&amp;1\\\\[1.1ex] 2&amp;4\\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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> We berekenen eerst de determinant van de matrix minus \u03bb op zijn hoofddiagonaal:<\/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-fadce42062bb04b7477318fdc35c4285_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}3- \\lambda &amp;1\\\\[1.1ex] 2&amp;4-\\lambda \\end{vmatrix} = \\lambda^2-7\\lambda +10\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"348\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Laten we nu de wortels van de karakteristieke polynoom berekenen:<\/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-7139127430fa6b78b78715d57a6fdf1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda^2-7\\lambda +10=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda = 2 \\\\[2ex] \\lambda = 5 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"239\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We berekenen de eigenvector die hoort bij de eigenwaarde 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-1b024c5f7e5acd0be55824c37befc587_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-614f9247b0d79635f70ec79eaa8c6529_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}1&amp;1\\\\[1.1ex] 2&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-272495fa6e8f89ba4e7c6a6d848cb38a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} x+y = 0 \\\\[2ex] 2x+2y = 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-77c240aaa8b75f1e5353c295ee86ad50_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\"> En dan berekenen we de eigenvector die hoort bij de eigenwaarde 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-b8aa7cae3057d78343128cd1095df24e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}-2&amp;1\\\\[1.1ex] 2&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-d8c38e500cf7103b1dc0e91ea1b4531a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -2x+y = 0 \\\\[2ex] 2x-y = 0\\end{array}\\right\\} \\longrightarrow \\ y=2x\" 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-8be56f81b5aef28783636f85c4dbd643_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] 2 \\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\"> Daarom zijn de eigenwaarden en eigenvectoren van matrix A: <\/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-cde889d89562f2e42bd6610b0045c118_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 2 \\qquad v = \\begin{pmatrix}1 \\\\[1.1ex] -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"158\" 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-3e954ba60fdc7eba60ba8530980854c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 5 \\qquad v = \\begin{pmatrix}1\\\\[1.1ex] 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"144\" 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<h3 class=\"wp-block-heading\"> Oefening 2<\/h3>\n<p> Bepaal de eigenwaarden en eigenvectoren van de volgende 2&#215;2 vierkante matrix: <\/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-54b0188c9fbadd6c3e35315443b71efd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}2&amp;1\\\\[1.1ex] 3&amp;0\\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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> We berekenen eerst de determinant van de matrix minus \u03bb op zijn hoofddiagonaal om de karakteristieke vergelijking te verkrijgen:<\/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-88fcd3b21ad2fa5a4d1d7789a86043e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}2- \\lambda &amp;1\\\\[1.1ex] 3&amp;-\\lambda \\end{vmatrix} = \\lambda^2-2\\lambda -3\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Laten we nu de wortels van de karakteristieke polynoom berekenen:<\/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-2614817b28bdb25c4fd89d4c773b4e35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda^2-2\\lambda -3=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda = -1 \\\\[2ex] \\lambda = 3 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"244\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We berekenen de eigenvector die hoort bij de eigenwaarde -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-f3e4bb6ba47bb4b9084b8a34d03dd35f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-(-1)I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"135\" 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-cdddf8c6da3e8066da62f60da7e9c603_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A+1I)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-e4b3d926f1a25454c3e645d79b28887d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 3&amp;1\\\\[1.1ex] 3&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=\"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-a7a6529e3ed8eb1607caa88475bcbb8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 3x+1y = 0 \\\\[2ex] 3x+1y = 0\\end{array}\\right\\} \\longrightarrow \\ y=-3x\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"225\" 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-67716508d5a9772f98c3f006f012dff1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] -3 \\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\"> En dan berekenen we de eigenvector die hoort bij de eigenwaarde 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-20e5d0be7e6dbe91bf15c835dac63b38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}-1&amp;1\\\\[1.1ex] 3&amp;-3\\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-8355b1ade79ba1508633f309926bc221_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -1x+1y = 0 \\\\[2ex] 3x-3y = 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\"> Daarom zijn de eigenwaarden en eigenvectoren van matrix A: <\/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-95e2b0bf0405bc0c301600cbb4b2b28a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = -1 \\qquad v = \\begin{pmatrix}1 \\\\[1.1ex] -3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"172\" 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-6322d97c5d24c1227b06dddf4b0974c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 3 \\qquad v = \\begin{pmatrix}1\\\\[1.1ex] 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"144\" 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\">Oefening 3<\/h3>\n<p> Bepaal de eigenwaarden en de eigenvectoren van de volgende orde 3-matrix: <\/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-a0e4f4147cbc9e0b657ff432f64bc8e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}1&amp;2&amp;0\\\\[1.1ex] 2&amp;1&amp;0\\\\[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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> We moeten eerst de determinant van matrix A minus de identiteitsmatrix, vermenigvuldigd met lambda, oplossen om de karakteristieke vergelijking te verkrijgen:<\/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-0af2ff4694103925883916b6a974c84d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}1-\\lambda&amp;2&amp;0\\\\[1.1ex] 2&amp;1-\\lambda&amp;0\\\\[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\"> In dit geval heeft de laatste kolom van de determinant twee nullen, dus we zullen hiervan profiteren om de determinant via cofactoren (of complementen) via deze kolom te berekenen:<\/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-ec7e7a2ec96b8d0721392c28838d105e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix}1-\\lambda&amp;2&amp;0\\\\[1.1ex] 2&amp;1-\\lambda&amp;0\\\\[1.1ex] 0&amp;1&amp;2-\\lambda\\end{vmatrix}&amp; = (2-\\lambda)\\cdot  \\begin{vmatrix}1-\\lambda&amp;2\\\\[1.1ex] 2&amp;1-\\lambda \\end{vmatrix} \\\\[3ex] &amp; = (2-\\lambda)[\\lambda^2 -2\\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\"> We moeten nu de wortels van de karakteristieke polynoom berekenen. Het is beter om de haakjes niet te vermenigvuldigen, omdat we dan een polynoom van de derde graad zouden krijgen. Aan de andere kant is het gemakkelijker om de eigenwaarden te verkrijgen als de twee factoren afzonderlijk worden opgelost:<\/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-adbfb1815d4a480c0584dfee1d8039fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (2-\\lambda)[\\lambda^2 -2\\lambda -3]=0 \\ \\longrightarrow \\ \\begin{cases} 2-\\lambda=0 \\ \\longrightarrow \\ \\lambda = 2 \\\\[2ex] \\lambda^2 -2\\lambda -3=0 \\ \\longrightarrow \\begin{cases}\\lambda = -1 \\\\[2ex] \\lambda = 3 \\end{cases} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"489\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We berekenen de eigenvector die hoort bij de eigenwaarde 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-a6a12c460df4d2f44709c4fd595193dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} -1&amp;2&amp;0\\\\[1.1ex] 2&amp;-1&amp;0\\\\[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-5c24e4b7b060a826203e3a049ddfc191_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -x+2y = 0 \\\\[2ex] 2x-y = 0\\\\[2ex] y=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} y=0 \\\\[2ex] x=y=0 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"248\" 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-75ebf6f61121b67afd80cdcec30a1709_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}0 \\\\[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\"> We berekenen de eigenvector die hoort bij de eigenwaarde -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-d23c4438a53032df27cc5334d4437c18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 2&amp;2&amp;0\\\\[1.1ex] 2&amp;2&amp;0\\\\[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=\"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-e023d57d34510e5e8f3a37c20d170e72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 2x+2y = 0 \\\\[2ex] 2x+2y = 0\\\\[2ex] y+3z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} x=-y \\\\[2ex] y=-3z \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"233\" 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-0be9ef18fb17845818bdd9de51dcb114_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}3 \\\\[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\"> We berekenen de eigenvector die hoort bij de eigenwaarde 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-d0360154b87545dd87e1b0b7bc06f4e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} -2&amp;2&amp;0\\\\[1.1ex] 2&amp;-2&amp;0\\\\[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=\"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-b8d514791286e43eae4b09d893d528df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -2x+2y = 0 \\\\[2ex] 2x-2y = 0\\\\[2ex] y-z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} x=y \\\\[2ex] y=z \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"224\" 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-99f86f65a5a9c69119285377d88f2efa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[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\"> Daarom zijn de eigenwaarden en eigenvectoren van matrix A: <\/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-fe42249314c1698847242c608bd65843_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 2 \\qquad v = \\begin{pmatrix}0 \\\\[1.1ex] 0 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"146\" 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-d412e1f81df9d6425db73113aaae5cd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = -1 \\qquad v = \\begin{pmatrix}3 \\\\[1.1ex] -3 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"174\" 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-f581aa37c9698dfb32062777a5a75b11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 3 \\qquad v = \\begin{pmatrix}1\\\\[1.1ex] 1 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"146\" 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\">Oefening 4<\/h3>\n<p> Bereken de eigenwaarden en eigenvectoren van de volgende 3&#215;3 vierkante matrix: <\/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-1323184f42d56f070e5b46a75a2e5c4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}2&amp;1&amp;3\\\\[1.1ex]-1&amp;1&amp;1\\\\[1.1ex] 1&amp;2&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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> We lossen eerst de determinant van de matrix minus \u03bb op zijn hoofddiagonaal op om de karakteristieke vergelijking te verkrijgen:<\/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-bc48c8489b25004ef131cc6ced36b929_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}2-\\lambda&amp;1&amp;3\\\\[1.1ex]-1&amp;1-\\lambda&amp;1\\\\[1.1ex] 1&amp;2&amp;4-\\lambda\\end{vmatrix}=-\\lambda^3+7\\lambda^2-10\\lambda\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"437\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We extraheren een gemeenschappelijke factor uit de karakteristieke polynoom en lossen \u03bb op uit elke vergelijking:<\/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-411dab2f65b426c37f8427d81ef13e97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda(-\\lambda^2+7\\lambda-10)=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda=0\\\\[2ex] -\\lambda^2+7\\lambda-10=0 \\ \\longrightarrow \\begin{cases}\\lambda = 2 \\\\[2ex] \\lambda = 5 \\end{cases} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"481\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We berekenen de eigenvector die hoort bij de eigenwaarde 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-bd7ebe2424c6524d522d5bba16d72d33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 2&amp;1&amp;3\\\\[1.1ex]-1&amp;1&amp;1\\\\[1.1ex] 1&amp;2&amp;4\\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-12cd10d2dc8afdb7a045beae4946b64d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 2x+y+3z= 0 \\\\[2ex] -x+y+z= 0\\\\[2ex] x+2y+4z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} x=-\\cfrac{2z}{3} \\\\[4ex] y=-\\cfrac{5z}{3} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"104\" width=\"266\" 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-a34877b285f281c83d7e73fa8eb40b9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}-2 \\\\[1.1ex] -5\\\\[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\"> We berekenen de eigenvector die hoort bij de eigenwaarde 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-fbcc697a3be877838fae3507dd3c1b68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 0&amp;1&amp;3\\\\[1.1ex]-1&amp;-1&amp;1\\\\[1.1ex] 1&amp;2&amp;2\\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-bb8d470bc7bff9f5d8d5a0245b1e7cbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} y+3z = 0 \\\\[2ex] -x-y+z= 0\\\\[2ex] x+2y+2z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} y=-3z \\\\[2ex] x=4z \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"263\" 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-30e589c5ae6b940b901454c296d8342b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}4\\\\[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\"> We berekenen de eigenvector die hoort bij de eigenwaarde 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-aaf6f17dedf5eecd1e035b9da59da2c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} -3&amp;1&amp;3\\\\[1.1ex]-1&amp;-4&amp;1\\\\[1.1ex] 1&amp;2&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\">\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dca8569528fb4923639dd535e25a0f74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -3x+y+3z = 0 \\\\[2ex] -x-4y+z = 0\\\\[2ex] x+2y-z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} x=z \\\\[2ex] y=0 \\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-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\"> Daarom zijn de eigenwaarden en eigenvectoren van matrix A: <\/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-62d8a98f007b72910fcd79622eda19e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 0 \\qquad v = \\begin{pmatrix}-2 \\\\[1.1ex] -5 \\\\[1.1ex] 3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"160\" 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-ee67e876a46b09430d2d73a653f2d743_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 2 \\qquad v = \\begin{pmatrix}4 \\\\[1.1ex] -3 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"160\" 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-99e4e8b0b837c26991777a294f30d49a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 5 \\qquad v = \\begin{pmatrix}1\\\\[1.1ex] 0 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"146\" 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-119\"><\/div>\n<\/div>\n<h3 class=\"wp-block-heading\"> Oefening 5<\/h3>\n<p> Bereken de eigenwaarden en eigenvectoren van de volgende 3&#215;3-matrix: <\/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-a39253beac54a05e9e84d431daf43362_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}2&amp;2&amp;2\\\\[1.1ex] 1&amp;2&amp;0\\\\[1.1ex] 0&amp;1&amp;3\\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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> We lossen eerst de determinant van de matrix minus \u03bb op zijn hoofddiagonaal op om de karakteristieke vergelijking te verkrijgen:<\/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-9392bbf957bee6c445c64192ae96a2ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}2-\\lambda&amp;2&amp;2\\\\[1.1ex] 1&amp;2-\\lambda&amp;0\\\\[1.1ex] 0&amp;1&amp;3-\\lambda\\end{vmatrix}=-\\lambda^3+7\\lambda^2-14\\lambda+8\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"468\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We vinden een wortel van het karakteristieke polynoom of het minimumpolynoom met behulp van de regel van 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-152ec29207fec8bdac7dabe9e1fbff31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{array}{r|rrrr} &amp; -1&amp;7&amp;-14&amp;8 \\\\[2ex] 1 &amp; &amp; -1&amp;6&amp;-8 \\\\ \\hline &amp;-1\\vphantom{\\Bigl)}&amp;6&amp;-8&amp;0 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"93\" width=\"190\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En dan vinden we de wortels van de verkregen polynoom:<\/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-b92304d107c097ec5712527929011440_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -\\lambda^2+6\\lambda -8=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda =2 \\\\[2ex] \\lambda = 4 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"244\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> De eigenwaarden van de matrix zijn dus:<\/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-05492792022e885b332adb0cbba45a0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda=1 \\qquad \\lambda =2 \\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\"> We berekenen de eigenvector die hoort bij de eigenwaarde 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-78173073be8dbdcab8a122ade043906d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-1I)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-981cc7881e44436326a35a7cc36ad26a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1&amp;2&amp;2\\\\[1.1ex] 1&amp;1&amp;0\\\\[1.1ex] 0&amp;1&amp;2\\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-d928870722dec65e8b48f7175d5dd4ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} x+2y+2z= 0 \\\\[2ex] x+y= 0\\\\[2ex] y+2z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} x=-y \\\\[2ex] y=-2z \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"263\" 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-da5ca9263773369d5824688b71a31644_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}2 \\\\[1.1ex] -2\\\\[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\"> We berekenen de eigenvector die hoort bij de eigenwaarde 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-b9d1686e2947a9bbe1dc10b373128e1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 0&amp;2&amp;2\\\\[1.1ex] 1&amp;0&amp;0\\\\[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-6000063fd1cc954e119cd5d73d08c405_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 2y+2z = 0 \\\\[2ex] x= 0\\\\[2ex] y+z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} y=-z \\\\[2ex] x=0\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"222\" 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-d47216be7fc08447ac3022a105a086b1_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\"> We berekenen de eigenvector die hoort bij de eigenwaarde 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-15e38e9899a9e8bb47cfbf10a4f05075_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} -2&amp;2&amp;2\\\\[1.1ex] 1&amp;-2&amp;0\\\\[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=\"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-004d61132ba8eeee123d8614432cbce2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -2x+2y+2z = 0 \\\\[2ex] x-2y = 0\\\\[2ex] y-z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} x=2y \\\\[2ex] y=z \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"273\" 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-5871bb6e88776aab87e0239540d43677_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=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Daarom zijn de eigenwaarden en eigenvectoren van matrix A: <\/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-f1ea8e2eff0c179b9872da8f6fab2d4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 1 \\qquad v = \\begin{pmatrix}2\\\\[1.1ex] -2 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"160\" 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-edc6fd09f9c6a12b26518a9103cc6610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 2 \\qquad v = \\begin{pmatrix}0 \\\\[1.1ex] -1 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"160\" 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-b492e98d771e76e77dc68d2fe2ea92c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 4 \\qquad v = \\begin{pmatrix}2 \\\\[1.1ex] 1 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"146\" 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\">Oefening 6<\/h3>\n<p> Zoek de eigenwaarden en eigenvectoren van de volgende 4\u00d74-matrix: <\/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-5cb04190d6f536d33b22265317441144_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix}1&amp;0&amp;-1&amp;0\\\\[1.1ex] 2&amp;-1&amp;-3&amp;0\\\\[1.1ex] -2&amp;0&amp;2&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"189\" 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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> We moeten eerst de determinant van de matrix minus \u03bb op zijn hoofddiagonaal oplossen om de karakteristieke vergelijking te verkrijgen:<\/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-35cae2dd143d77e22a522b49e8d43f3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}1-\\lambda&amp;0&amp;-1&amp;0\\\\[1.1ex] 2&amp;-1-\\lambda&amp;-3&amp;0\\\\[1.1ex] -2&amp;0&amp;2-\\lambda&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;3-\\lambda\\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"352\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> In dit geval bevat de laatste kolom van de determinant alleen nullen behalve \u00e9\u00e9n element. We zullen hier daarom van profiteren om de determinant via cofactoren via deze kolom te berekenen:<\/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-456b0612b308c03fd1643a5ba0f332e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix}1-\\lambda&amp;0&amp;-1&amp;0\\\\[1.1ex] 2&amp;-1-\\lambda&amp;-3&amp;0\\\\[1.1ex] -2&amp;0&amp;2-\\lambda&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;3-\\lambda\\end{vmatrix}&amp; = (3-\\lambda)\\cdot  \\begin{vmatrix}1-\\lambda&amp;0&amp;-1\\\\[1.1ex] 2&amp;-1-\\lambda&amp;-3\\\\[1.1ex] -2&amp;0&amp;2-\\lambda\\end{vmatrix} \\\\[3ex] &amp; = (3-\\lambda)[-\\lambda^3 +2\\lambda^2 +3\\lambda] \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"161\" width=\"505\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We moeten nu de wortels van de karakteristieke polynoom berekenen. Het is beter om de haakjes niet te vermenigvuldigen, omdat we dan een polynoom van de vierde graad zouden krijgen, maar als de twee factoren afzonderlijk worden opgelost, is het gemakkelijker om de eigenwaarden te berekenen: <\/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-ef6e59f8631cac087c988004aa512b62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (3-\\lambda)[-\\lambda^3 +2\\lambda^2 +3\\lambda]=0 \\ \\longrightarrow \\ \\begin{cases} 3-\\lambda=0 \\ \\longrightarrow \\ \\lambda = 3 \\\\[2ex] -\\lambda^3 +2\\lambda^2 +3\\lambda =0 \\ \\longrightarrow \\ \\lambda(-\\lambda^2 +2\\lambda +3) =0 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"620\" 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-786b2892e7045f117498697407d35552_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda(-\\lambda^2 +2\\lambda +3)=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda=0  \\\\[2ex] -\\lambda^2 +2\\lambda +3=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda=-1 \\\\[2ex] \\lambda = 3 \\end{cases}\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"483\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We berekenen de eigenvector die hoort bij de eigenwaarde 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-f43f22947b29779ef456e4ac7a5d66a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1&amp;0&amp;-1&amp;0\\\\[1.1ex] 2&amp;-1&amp;-3&amp;0\\\\[1.1ex] -2&amp;0&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=\"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-c1d7e96203dceb7288f89ab932532351_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} w-y = 0 \\\\[2ex] 2w-x-3y = 0\\\\[2ex] -2w+2y=0 \\\\[2ex] 3z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} w=y \\\\[2ex] x=-w  \\\\[2ex]z=0 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"263\" 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-3c4a8b3ef3502a2bf8efd6cc398b5ae6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] -1 \\\\[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\"> We berekenen de eigenvector die hoort bij de eigenwaarde -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-cdddf8c6da3e8066da62f60da7e9c603_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A+1I)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-7fbbdceca419f15672da0dcb7c15078c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 2&amp;0&amp;-1&amp;0\\\\[1.1ex] 2&amp;0&amp;-3&amp;0\\\\[1.1ex] -2&amp;0&amp;3&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;4\\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-3ff9a5afa0cefa73985e7ba00c945dac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 2w-y = 0 \\\\[2ex] 2w-3y = 0\\\\[2ex] -2w+3y=0 \\\\[2ex] 4z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} y=w=0  \\\\[2ex]z=0 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"263\" 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-daee73fdcebacce8a5e5f7104ed9c213_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}0 \\\\[1.1ex] 1 \\\\[1.1ex] 0  \\\\[1.1ex]0 \\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\"> We berekenen de eigenvector die hoort bij de eigenwaarde 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-6493c6019a8b9be3254db2ffeaa19703_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} -2&amp;0&amp;-1&amp;0\\\\[1.1ex] 2&amp;-4&amp;-3&amp;0\\\\[1.1ex] -2&amp;0&amp;-1&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-fbcbd01420d80be317ecbec57010b662_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -2w-y = 0 \\\\[2ex] 2w-4x-3y = 0\\\\[2ex] -2w-y=0 \\\\[2ex] 0=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} y=-2w \\\\[2ex] x=2w  \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"280\" 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-3b70eeb51bea073f058763401adf5240_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] 2 \\\\[1.1ex] -2  \\\\[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\"> De eigenwaarde 3 heeft een veelvoud gelijk aan 2, omdat deze tweemaal wordt herhaald. We moeten daarom een andere eigenvector vinden die aan dezelfde vergelijkingen voldoet:<\/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-5dc5fd38503b7683d8a7e3df9da9ee8d_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\"> Daarom zijn de eigenwaarden en eigenvectoren van matrix A: <\/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-b8bd7188d1d3ed1abe178d9b5f5bbc0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 0 \\qquad v = \\begin{pmatrix}1 \\\\[1.1ex] -1 \\\\[1.1ex] 1  \\\\[1.1ex]0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"160\" 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-2ea128d2a6e5387bd538ac3d0119b2ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = -1 \\qquad v = \\begin{pmatrix}0 \\\\[1.1ex] 1 \\\\[1.1ex] 0  \\\\[1.1ex]0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"160\" 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-22d83a8f13bdb44bf1c23f3c6b963d65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 3 \\qquad v = \\begin{pmatrix}1 \\\\[1.1ex] 2 \\\\[1.1ex] -2  \\\\[1.1ex]0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"160\" 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-65cd815fe71a1c6d8063f0f78e3422a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 3 \\qquad v = \\begin{pmatrix}0 \\\\[1.1ex] 0 \\\\[1.1ex] 0  \\\\[1.1ex]1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"146\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Op deze pagina leggen we uit wat eigenwaarden en eigenvectoren zijn, ook wel eigenwaarden en eigenvectoren genoemd. U vindt er ook voorbeelden van hoe u ze kunt berekenen, evenals stapsgewijze opgeloste oefeningen om te oefenen. Wat is een eigenwaarde en een eigenvector? Hoewel het begrip eigenwaarde en eigenvector moeilijk te begrijpen is, is de definitie &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/eigenwaarden-berekenen-eigenwaarden-en-eigenvectoren-eigenvectoren-van-een-matrix\/\"> <span class=\"screen-reader-text\">Eigenwaarden (of eigenwaarden) en eigenvectoren (of eigenvectoren) van een matrix<\/span> Lees meer &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-375","post","type-post","status-publish","format-standard","hentry","category-determinant-van-een-matrix"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Eigenwaarden (of eigenwaarden) en eigenvectoren (of eigenvectoren) van een matrix -<\/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\/nl\/eigenwaarden-berekenen-eigenwaarden-en-eigenvectoren-eigenvectoren-van-een-matrix\/\" \/>\n<meta property=\"og:locale\" content=\"nl_NL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Eigenwaarden (of eigenwaarden) en eigenvectoren (of eigenvectoren) van een matrix -\" \/>\n<meta property=\"og:description\" content=\"Op deze pagina leggen we uit wat eigenwaarden en eigenvectoren zijn, ook wel eigenwaarden en eigenvectoren genoemd. 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Hoewel het begrip eigenwaarde en eigenvector moeilijk te begrijpen is, is de definitie &hellip; Eigenwaarden (of eigenwaarden) en eigenvectoren (of eigenvectoren) van een matrix Lees meer &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/nl\/eigenwaarden-berekenen-eigenwaarden-en-eigenvectoren-eigenvectoren-van-een-matrix\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T06:35:36+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-710a5e2df8739c35c060f790f5592734_l3.png\" \/>\n<meta name=\"author\" content=\"Redactioneel Team\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Geschreven door\" \/>\n\t<meta name=\"twitter:data1\" content=\"Redactioneel Team\" \/>\n\t<meta name=\"twitter:label2\" content=\"Geschatte leestijd\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuut\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/nl\/eigenwaarden-berekenen-eigenwaarden-en-eigenvectoren-eigenvectoren-van-een-matrix\/\",\"url\":\"https:\/\/mathority.org\/nl\/eigenwaarden-berekenen-eigenwaarden-en-eigenvectoren-eigenvectoren-van-een-matrix\/\",\"name\":\"Eigenwaarden (of eigenwaarden) en eigenvectoren (of eigenvectoren) van een matrix -\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/nl\/#website\"},\"datePublished\":\"2023-07-06T06:35:36+00:00\",\"dateModified\":\"2023-07-06T06:35:36+00:00\",\"author\":{\"@id\":\"https:\/\/mathority.org\/nl\/#\/schema\/person\/19b550cef1a9fbd238be112b7b7bbf64\"},\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/nl\/eigenwaarden-berekenen-eigenwaarden-en-eigenvectoren-eigenvectoren-van-een-matrix\/#breadcrumb\"},\"inLanguage\":\"nl-NL\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/nl\/eigenwaarden-berekenen-eigenwaarden-en-eigenvectoren-eigenvectoren-van-een-matrix\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/nl\/eigenwaarden-berekenen-eigenwaarden-en-eigenvectoren-eigenvectoren-van-een-matrix\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/nl\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Eigenwaarden (of eigenwaarden) en eigenvectoren (of eigenvectoren) van een matrix\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathority.org\/nl\/#website\",\"url\":\"https:\/\/mathority.org\/nl\/\",\"name\":\"\",\"description\":\"Waar nieuwsgierigheid en berekening elkaar ontmoeten!\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathority.org\/nl\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"nl-NL\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathority.org\/nl\/#\/schema\/person\/19b550cef1a9fbd238be112b7b7bbf64\",\"name\":\"Redactioneel Team\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"nl-NL\",\"@id\":\"https:\/\/mathority.org\/nl\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"caption\":\"Redactioneel Team\"},\"sameAs\":[\"http:\/\/mathority.org\/nl\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Eigenwaarden (of eigenwaarden) en eigenvectoren (of eigenvectoren) van een matrix -","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/nl\/eigenwaarden-berekenen-eigenwaarden-en-eigenvectoren-eigenvectoren-van-een-matrix\/","og_locale":"nl_NL","og_type":"article","og_title":"Eigenwaarden (of eigenwaarden) en eigenvectoren (of eigenvectoren) van een matrix -","og_description":"Op deze pagina leggen we uit wat eigenwaarden en eigenvectoren zijn, ook wel eigenwaarden en eigenvectoren genoemd. 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