{"id":317,"date":"2023-07-06T10:46:31","date_gmt":"2023-07-06T10:46:31","guid":{"rendered":"https:\/\/mathority.org\/it\/matrice-normale\/"},"modified":"2023-07-06T10:46:31","modified_gmt":"2023-07-06T10:46:31","slug":"matrice-normale","status":"publish","type":"post","link":"https:\/\/mathority.org\/it\/matrice-normale\/","title":{"rendered":"Matrice regolare"},"content":{"rendered":"<p>In questa pagina vedrai cos&#8217;\u00e8 una matrice normale e alcuni esempi di matrici normali. Inoltre troverai le propriet\u00e0 di questo tipo di matrici e gli esercizi risolti passo dopo passo.<\/p>\n<h2 class=\"wp-block-heading\"> Cos&#8217;\u00e8 una matrice normale?<\/h2>\n<p> La definizione di array normale \u00e8: <\/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\"> Una <strong>matrice normale<\/strong> \u00e8 una matrice complessa che moltiplicata per la sua <a href=\"https:\/\/mathority.org\/it\/coniugato-di-matrice-complessa-e-coniugato-di-trasposizione\/\">matrice di trasposizione coniugata<\/a> \u00e8 uguale al prodotto della trasposizione coniugata da sola.<\/p>\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\"> Oro<\/p>\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> \u00e8 la matrice di trasposizione coniugata di<\/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> Se per\u00f2 si tratta di matrici <strong>di numeri reali<\/strong> , la condizione precedente equivale a dire che una matrice commuta con la sua trasposta, cio\u00e8:<\/p>\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> Perch\u00e9, ovviamente, la matrice di trasposizione coniugata di una matrice reale \u00e8 semplicemente la matrice di trasposizione (o trasposizione).<\/p>\n<h2 class=\"wp-block-heading\"> Esempi di matrici normali<\/h2>\n<h3 class=\"wp-block-heading\"> Esempio con numeri complessi<\/h3>\n<p> La seguente matrice quadrata complessa di dimensione 2\u00d72 \u00e8 normale: <\/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=\"esempio di matrice normale con numeri complessi di dimensione 2x2\" class=\"wp-image-2041\" width=\"125\" height=\"65\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> La dimostrazione della sua normalit\u00e0 \u00e8 allegata di seguito:<\/p>\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\"> Esempio con numeri reali<\/h3>\n<p> Anche la seguente matrice quadrata con numeri reali di ordine 2 \u00e8 normale: <\/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=\"esempio di matrice normale con numeri reali di dimensione 2x2\" class=\"wp-image-2042\" width=\"130\" height=\"68\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> In questo caso, poich\u00e9 ha solo numeri reali, per dimostrare che \u00e8 normale \u00e8 sufficiente verificare che la matrice sia commutabile con la sua trasposta:<\/p>\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\"> Propriet\u00e0 delle matrici normali<\/h2>\n<p> Le matrici normali hanno le seguenti caratteristiche:<\/p>\n<ul>\n<li> Tutte le matrici normali sono matrici diagonalizzabili.<\/li>\n<\/ul>\n<ul>\n<li> Ogni <a href=\"https:\/\/mathority.org\/it\/matrice-unitaria\/\">matrice unitaria<\/a> \u00e8 anche una matrice normale.<\/li>\n<\/ul>\n<ul>\n<li> Allo stesso modo, una <a href=\"https:\/\/mathority.org\/it\/matrice-hermitiana-o-hermitiana\/\">matrice hermitiana<\/a> \u00e8 una matrice normale.<\/li>\n<\/ul>\n<ul>\n<li> Allo stesso modo, una matrice antihermitiana \u00e8 una matrice normale.<\/li>\n<\/ul>\n<ul>\n<li> Se A \u00e8 una matrice normale, gli autovalori (o autovalori) della matrice trasposta coniugata A* sono gli autovalori coniugati di 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> Nelle matrici normali gli autovettori (o autovettori) associati ai diversi autovalori sono ortogonali.<\/li>\n<\/ul>\n<ul>\n<li> Se una matrice \u00e8 composta solo da numeri reali ed \u00e8 <a href=\"https:\/\/mathority.org\/it\/esempi-e-proprieta-di-matrici-simmetriche\/\">simmetrica<\/a> , \u00e8 allo stesso tempo una matrice normale.<\/li>\n<\/ul>\n<ul>\n<li> Allo stesso modo, anche una <a href=\"https:\/\/mathority.org\/it\/esempi-e-proprieta-di-matrici-antisimmetriche\/\">matrice reale antisimmetrica<\/a> \u00e8 una matrice normale.<\/li>\n<\/ul>\n<ul>\n<li> Infine, anche qualsiasi matrice ortogonale formata da numeri reali \u00e8 una matrice normale.<\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"> Esercizi risolti per matrici normali<\/h2>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<h3 class=\"wp-block-heading\"> Esercizio 1<\/h3>\n<p> Verificare che la seguente matrice complessa di dimensione 2 \u00d7 2 sia normale: <\/p>\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>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Per dimostrare che la matrice \u00e8 normale dobbiamo prima calcolare la sua trasposta coniugata:<\/p>\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\"> E ora facciamo la verifica moltiplicando la matrice A per la matrice A* in entrambe le possibili direzioni: <\/p>\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\"> Il risultato di entrambe le moltiplicazioni \u00e8 lo stesso, quindi <strong>la matrice A \u00e8 normale.<\/strong><\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Esercizio 2<\/h3>\n<p> Mostra che la seguente matrice reale di dimensione 2 \u00d7 2 \u00e8 normale: <\/p>\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>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Poich\u00e9 in questo caso abbiamo a che fare con un ambiente di soli numeri reali, \u00e8 sufficiente verificare che il prodotto matriciale tra la matrice A e la sua trasposta d\u00e0 lo stesso risultato qualunque sia la direzione della moltiplicazione: <\/p>\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\"> Il risultato di entrambi i prodotti \u00e8 lo stesso, quindi <strong>la matrice A \u00e8 normale.<\/strong><\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Esercizio 3<\/h3>\n<p> Determina se la seguente matrice di numeri complessi di ordine 2 \u00e8 normale: <\/p>\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>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Per verificare che la matrice sia normale, dobbiamo prima calcolare la sua trasposta coniugata:<\/p>\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\"> E ora controlliamo se la matrice A e la sua trasposizione coniugata sono commutabili: <\/p>\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\"> Il risultato di entrambe le moltiplicazioni \u00e8 lo stesso, quindi <strong>la matrice A \u00e8 normale.<\/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\"> Esercizio 4<\/h3>\n<p> Verificare che la seguente matrice reale di dimensione 3\u00d73 \u00e8 normale: <\/p>\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>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Essendo la matrice interamente composta da elementi reali, \u00e8 sufficiente verificare che il prodotto matriciale tra la matrice A e la sua trasposta \u00e8 indipendente dalla direzione della moltiplicazione: <\/p>\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\"> Il risultato di entrambi i prodotti \u00e8 lo stesso, quindi <strong>la matrice A \u00e8 normale.<\/strong><\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Esercizio 5<\/h3>\n<p> Determina se la seguente matrice complessa di ordine 3\u00d73 \u00e8 normale: <\/p>\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>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Innanzitutto, calcoliamo la trasposta coniugata della matrice:<\/p>\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\"> Ora dobbiamo fare le moltiplicazioni di matrice tra la matrice A e il suo coniugato trasposto in entrambe le possibili direzioni. Tuttavia, la matrice trasposta coniugata di A \u00e8 uguale alla matrice A stessa, quindi \u00e8 una matrice hermitiana. E quindi, <strong>dalle propriet\u00e0 delle matrici normali ne consegue che A \u00e8 una matrice normale<\/strong> , perch\u00e9 ogni matrice Hermitiana \u00e8 una matrice normale.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>In questa pagina vedrai cos&#8217;\u00e8 una matrice normale e alcuni esempi di matrici normali. Inoltre troverai le propriet\u00e0 di questo tipo di matrici e gli esercizi risolti passo dopo passo. Cos&#8217;\u00e8 una matrice normale? La definizione di array normale \u00e8: Una matrice normale \u00e8 una matrice complessa che moltiplicata per la sua matrice di trasposizione &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/it\/matrice-normale\/\"> <span class=\"screen-reader-text\">Matrice regolare<\/span> Leggi altro &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":[32],"tags":[],"class_list":["post-317","post","type-post","status-publish","format-standard","hentry","category-tipi-di-tabelle"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Matrice regolare - 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\/it\/matrice-normale\/\" \/>\n<meta property=\"og:locale\" content=\"it_IT\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Matrice regolare - Mathority\" \/>\n<meta property=\"og:description\" content=\"In questa pagina vedrai cos&#8217;\u00e8 una matrice normale e alcuni esempi di matrici normali. 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