{"id":359,"date":"2023-07-06T10:46:31","date_gmt":"2023-07-06T10:46:31","guid":{"rendered":"https:\/\/mathority.org\/nl\/normale-matrix\/"},"modified":"2023-07-06T10:46:31","modified_gmt":"2023-07-06T10:46:31","slug":"normale-matrix","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/normale-matrix\/","title":{"rendered":"Reguliere matrix"},"content":{"rendered":"<p>Op deze pagina ziet u wat een normale matrix is en voorbeelden van normale matrices. Daarnaast vind je stap voor stap de eigenschappen van dit soort matrices en oefeningen opgelost.<\/p>\n<h2 class=\"wp-block-heading\"> Wat is een normale matrix?<\/h2>\n<p> De normale arraydefinitie is: <\/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 align=\"LEFT\"> Een <strong>normale matrix<\/strong> is een complexe matrix die, vermenigvuldigd met zijn <a href=\"https:\/\/mathority.org\/nl\/complex-matrixconjugaat-en-transponeerconjugaat\/\">geconjugeerde transpositiematrix,<\/a> gelijk is aan het product van de geconjugeerde transpositie op zichzelf.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-196deb39b1de9764cb4013ded78fe671_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\\cdot A^*=A^*\\cdot A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"118\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p 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-0d4c81a666954cf4d9d7889c69274641_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^*\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"19\" style=\"vertical-align: 0px;\"><\/p>\n<p> is de geconjugeerde getransponeerde matrix van<\/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> Als het echter <strong>re\u00eble<\/strong> getallenmatrices zijn, komt de voorgaande voorwaarde erop neer dat een matrix pendelt met zijn transpositie, dat wil zeggen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f808080cda647c3e7cbf2cac2129539_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\\cdot A^t=A^t\\cdot A\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"114\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Omdat de geconjugeerde transponeermatrix van een echte matrix uiteraard eenvoudigweg de transponeer- (of transponeer-) matrix is.<\/p>\n<h2 class=\"wp-block-heading\"> Voorbeelden van normale matrices<\/h2>\n<h3 class=\"wp-block-heading\"> Voorbeeld met complexe getallen<\/h3>\n<p> De volgende complexe vierkante matrix met afmeting 2\u00d72 is normaal: <\/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-matrice-normale-complexe-22152-1.webp\" alt=\"voorbeeld van een normale matrix met complexe getallen met dimensie 2x2\" class=\"wp-image-2041\" width=\"125\" height=\"65\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> De demonstratie van de normaliteit ervan is hieronder bijgevoegd:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f44b98cec879a8332c462d2393fbfbba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^* = \\begin{pmatrix} i &amp; i \\\\[1.1ex] i &amp; -i \\end{pmatrix} \\cdot \\begin{pmatrix} -i &amp; -i \\\\[1.1ex] -i &amp; i \\end{pmatrix} =\\begin{pmatrix} 2 &amp; 0 \\\\[1.1ex] 0 &amp; 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"319\" 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-fddc406493ac1c81c86edf1ad6e58d0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^*\\cdot A = \\begin{pmatrix} -i &amp; -i \\\\[1.1ex] -i &amp; i \\end{pmatrix}\\cdot \\begin{pmatrix} i &amp; i \\\\[1.1ex] i &amp; -i \\end{pmatrix}  = \\begin{pmatrix} 2 &amp; 0 \\\\[1.1ex] 0 &amp; 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"319\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Voorbeeld met re\u00eble getallen<\/h3>\n<p> De volgende vierkante matrix met re\u00eble getallen van orde 2 is ook normaal: <\/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\/example-real-normal-matrix-22152-1.webp\" alt=\"voorbeeld van een normale matrix met re\u00eble getallen met dimensie 2x2\" class=\"wp-image-2042\" width=\"130\" height=\"68\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> In dit geval, omdat het alleen re\u00eble getallen heeft, is het voldoende om te bewijzen dat het normaal is om te verifi\u00ebren dat de matrix commuteerbaar is met zijn transpositie:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a320a8e300315c6a48bb8095266408ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B\\cdot B^t = \\begin{pmatrix} 2 &amp; -2 \\\\[1.1ex] 2 &amp; 2 \\end{pmatrix} \\cdot \\begin{pmatrix} 2 &amp; 2 \\\\[1.1ex] -2 &amp; 2 \\end{pmatrix} =\\begin{pmatrix} 8 &amp; 0 \\\\[1.1ex] 0 &amp; 8 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"317\" 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-b6ad5bd62deeb5bcbf561a2ee6b29741_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B^t\\cdot B =\\begin{pmatrix} 2 &amp; 2 \\\\[1.1ex] -2 &amp; 2 \\end{pmatrix}\\cdot \\begin{pmatrix} 2 &amp; -2 \\\\[1.1ex] 2 &amp; 2 \\end{pmatrix} =\\begin{pmatrix} 8 &amp; 0 \\\\[1.1ex] 0 &amp; 8 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"317\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Eigenschappen van normale matrices<\/h2>\n<p> Normale matrices hebben de volgende kenmerken:<\/p>\n<ul>\n<li> Alle normale matrices zijn diagonaliseerbare matrices.<\/li>\n<\/ul>\n<ul>\n<li> Elke <a href=\"https:\/\/mathority.org\/nl\/unitaire-matrix\/\">unitaire matrix<\/a> is ook een normale matrix.<\/li>\n<\/ul>\n<ul>\n<li> Op dezelfde manier is een <a href=\"https:\/\/mathority.org\/nl\/hermitische-of-hermitische-matrix\/\">Hermitische matrix<\/a> een normale matrix.<\/li>\n<\/ul>\n<ul>\n<li> Op dezelfde manier is een antihermitische matrix een normale matrix.<\/li>\n<\/ul>\n<ul>\n<li> Als A een normale matrix is, zijn de eigenwaarden (of eigenwaarden) van de geconjugeerde getransponeerde matrix A* de geconjugeerde eigenwaarden van A.<\/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-a91ee46b5f8dda0d51ecb57474f5b816_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix}2i&amp;-1+i\\\\[1.1ex] 1+i&amp;i\\end{pmatrix} \\longrightarrow \\ \\lambda_{A,1} = 0 \\ ; \\ \\lambda_{A,2} = +3i\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"382\" 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-48c80a017a9afd8b4cf3923757f4e945_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^*=\\begin{pmatrix}-2i&amp;1-i\\\\[1.1ex] -1-i&amp;-i\\end{pmatrix} \\longrightarrow \\ \\lambda_{A^*,1} = 0 \\ ; \\ \\lambda_{A^*,2} = -3i\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"403\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> In normale matrices zijn de eigenvectoren (of eigenvectoren) die bij de verschillende eigenwaarden horen orthogonaal.<\/li>\n<\/ul>\n<ul>\n<li> Als een matrix alleen uit re\u00eble getallen bestaat en <a href=\"https:\/\/mathority.org\/nl\/symmetrische-matrixvoorbeelden-en-eigenschappen\/\">symmetrisch<\/a> is, is het tegelijkertijd een normale matrix.<\/li>\n<\/ul>\n<ul>\n<li> Op dezelfde manier is een <a href=\"https:\/\/mathority.org\/nl\/antisymmetrische-matrixvoorbeelden-en-eigenschappen\/\">antisymmetrische re\u00eble matrix<\/a> ook een normale matrix.<\/li>\n<\/ul>\n<ul>\n<li> Ten slotte is elke orthogonale matrix die uit re\u00eble getallen bestaat, ook een normale matrix.<\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"> Opgeloste oefeningen voor normale matrices<\/h2>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<h3 class=\"wp-block-heading\"> Oefening 1<\/h3>\n<p> Controleer of de volgende complexe matrix met afmeting 2 \u00d7 2 normaal is: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ff27d19373c5a4dc8e95472ec295c657_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix}1&amp;2+3i\\\\[1.1ex] 2+3i&amp;1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"168\" 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\"> Om aan te tonen dat de matrix normaal is, moeten we eerst de geconjugeerde transpositie 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-17c96c654ce5b978f90a905b973d5ae7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^*=\\begin{pmatrix}1&amp;2-3i\\\\[1.1ex] 2-3i&amp;1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"176\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En nu doen we de verificatie door matrix A te vermenigvuldigen met matrix A* in beide mogelijke richtingen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36212e1d12cf35ea5dd27bd91d77ee56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^* = \\begin{pmatrix}1&amp;2+3i\\\\[1.1ex] 2+3i&amp;1\\end{pmatrix}\\cdot \\begin{pmatrix}1&amp;2-3i\\\\[1.1ex] 2-3i&amp;1\\end{pmatrix} = \\begin{pmatrix}14&amp;4\\\\[1.1ex] 4&amp;14\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"453\" 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-3db0fc8fdc948037452b4c6275896686_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^*\\cdot A =\\begin{pmatrix}1&amp;2-3i\\\\[1.1ex] 2-3i&amp;1\\end{pmatrix}\\cdot  \\begin{pmatrix}1&amp;2+3i\\\\[1.1ex] 2+3i&amp;1\\end{pmatrix} = \\begin{pmatrix}14&amp;4\\\\[1.1ex] 4&amp;14\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"453\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Het resultaat van beide vermenigvuldigingen is hetzelfde, dus <strong>matrix A is normaal.<\/strong><\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Oefening 2<\/h3>\n<p> Laat zien dat de volgende re\u00eble matrix van maat 2 \u00d7 2 normaal is: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-854e13859be417985691b5ed6d2a050f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix}3&amp;5\\\\[1.1ex] -5&amp;3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"109\" 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\"> Omdat we in dit geval te maken hebben met een omgeving met alleen re\u00eble getallen, volstaat het om te verifi\u00ebren dat het matrixproduct tussen de matrix A en zijn transpositie hetzelfde resultaat geeft, ongeacht de richting van de vermenigvuldiging: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b1b6314188f394b3053d3dac0613cf5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^t = \\begin{pmatrix}3&amp;5\\\\[1.1ex] -5&amp;3\\end{pmatrix}\\cdot \\begin{pmatrix}3&amp;-5\\\\[1.1ex] 5&amp;3\\end{pmatrix} = \\begin{pmatrix}34&amp;0\\\\[1.1ex] 0&amp;34\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"332\" 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-e2b33f892cd29c0ee232b88eaa4946cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^t\\cdot A = \\begin{pmatrix}3&amp;-5\\\\[1.1ex] 5&amp;3\\end{pmatrix}\\cdot \\begin{pmatrix}3&amp;5\\\\[1.1ex] -5&amp;3\\end{pmatrix} = \\begin{pmatrix}34&amp;0\\\\[1.1ex] 0&amp;34\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"332\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Het resultaat van beide producten is hetzelfde, dus <strong>matrix A is normaal.<\/strong><\/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 of de volgende matrix van complexe getallen van orde 2 normaal is: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-00075db37b045e08349f7d5b3f679570_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix}4i&amp;-1+i\\\\[1.1ex] 1-i&amp;4i\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"164\" 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\"> Om te controleren of de matrix normaal is, moeten we eerst de geconjugeerde transpositie 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-0b39733376eb2aef269012eb1d6c24be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^*=\\begin{pmatrix}-4i&amp;1+i\\\\[1.1ex] -1-i&amp;-4i\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"172\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En nu controleren we of de matrix A en zijn geconjugeerde transpositie schakelbaar zijn: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c207cb9842dacbaf9bc59d4aaff00473_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^* = \\begin{pmatrix}4i&amp;-1+i\\\\[1.1ex] 1-i&amp;4i\\end{pmatrix}\\cdot \\begin{pmatrix}-4i&amp;1+i\\\\[1.1ex] -1-i&amp;-4i\\end{pmatrix} = \\begin{pmatrix}18&amp;8i\\\\[1.1ex] -8i&amp;18\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"456\" 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-bcf52f3da81fd7c56b090604c2b6f368_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^*\\cdot A =\\begin{pmatrix}-4i&amp;1+i\\\\[1.1ex] -1-i&amp;-4i\\end{pmatrix}\\cdot  \\begin{pmatrix}4i&amp;-1+i\\\\[1.1ex] 1-i&amp;4i\\end{pmatrix} = \\begin{pmatrix}18&amp;8i\\\\[1.1ex] -8i&amp;18\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"456\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Het resultaat van beide vermenigvuldigingen is hetzelfde, dus <strong>matrix A is normaal.<\/strong> <\/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 4<\/h3>\n<p> Controleer of de volgende re\u00eble matrix van dimensie 3\u00d73 normaal is: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-92ee07759c3e6e88af5a68479b5833ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} -1&amp;1&amp;0\\\\[1.1ex] 0&amp;-1&amp;1\\\\[1.1ex] 1&amp;0&amp;-1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"164\" 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\"> Omdat de matrix volledig uit re\u00eble elementen bestaat, volstaat het om te verifi\u00ebren dat het matrixproduct tussen de matrix A en zijn transpositie onafhankelijk is van de richting van de vermenigvuldiging: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dc7ee02c75239b430c7fc2418f43e343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^t = \\begin{pmatrix} -1&amp;1&amp;0\\\\[1.1ex] 0&amp;-1&amp;1\\\\[1.1ex] 1&amp;0&amp;-1\\end{pmatrix} \\cdot\\begin{pmatrix}-1&amp;0&amp;1\\\\[1.1ex] 1&amp;-1&amp;0\\\\[1.1ex] 0&amp;1&amp;-1\\end{pmatrix}=\\begin{pmatrix}2&amp;-1&amp;-1\\\\[1.1ex] -1&amp;2&amp;-1\\\\[1.1ex] -1&amp;-1&amp;2\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"495\" 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-e661b877ee225983c797584e2b61d429_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^t\\cdot A =\\begin{pmatrix}-1&amp;0&amp;1\\\\[1.1ex] 1&amp;-1&amp;0\\\\[1.1ex] 0&amp;1&amp;-1\\end{pmatrix}\\cdot \\begin{pmatrix} -1&amp;1&amp;0\\\\[1.1ex] 0&amp;-1&amp;1\\\\[1.1ex] 1&amp;0&amp;-1\\end{pmatrix}=\\begin{pmatrix}2&amp;-1&amp;-1\\\\[1.1ex] -1&amp;2&amp;-1\\\\[1.1ex] -1&amp;-1&amp;2\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"495\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Het resultaat van beide producten is hetzelfde, dus <strong>matrix A is normaal.<\/strong><\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Oefening 5<\/h3>\n<p> Bepaal of de volgende complexe matrix van orde 3\u00d73 normaal is: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-81ca0ac1da07c151a62dcfb06b4be877_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix}4&amp;3-2i &amp; 5i \\\\[1.1ex] 3+2i &amp; 0 &amp; -1-3i \\\\[1.1ex] -5i &amp; -1+3i &amp; 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"260\" 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\"> Eerst berekenen we de geconjugeerde transpositie 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-fd0a2dfe1b8bfe18020ab68c1eb3bda6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^*=\\begin{pmatrix}4&amp;3-2i &amp; 5i \\\\[1.1ex] 3+2i &amp; 0 &amp; -1-3i \\\\[1.1ex] -5i &amp; -1+3i &amp; 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"268\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu moeten we de matrixvermenigvuldigingen uitvoeren tussen matrix A en zijn geconjugeerde transpositie in beide mogelijke richtingen. De geconjugeerde transponeermatrix van A is echter gelijk aan de matrix A zelf, dus het is een Hermitische matrix. En daarom <strong>volgt uit de eigenschappen van normale matrices dat A een normale matrix is<\/strong> , omdat elke Hermitische matrix een normale matrix is.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Op deze pagina ziet u wat een normale matrix is en voorbeelden van normale matrices. Daarnaast vind je stap voor stap de eigenschappen van dit soort matrices en oefeningen opgelost. Wat is een normale matrix? De normale arraydefinitie is: Een normale matrix is een complexe matrix die, vermenigvuldigd met zijn geconjugeerde transpositiematrix, gelijk is aan &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/normale-matrix\/\"> <span class=\"screen-reader-text\">Reguliere 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":[64],"tags":[],"class_list":["post-359","post","type-post","status-publish","format-standard","hentry","category-soorten-tafels"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Reguliere matrix - 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\/nl\/normale-matrix\/\" \/>\n<meta property=\"og:locale\" content=\"nl_NL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Reguliere matrix - Mathority\" \/>\n<meta property=\"og:description\" content=\"Op deze pagina ziet u wat een normale matrix is en voorbeelden van normale matrices. 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