{"id":187636,"date":"2025-02-05T15:33:59","date_gmt":"2025-02-05T15:33:59","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=187636"},"modified":"2025-02-05T15:34:01","modified_gmt":"2025-02-05T15:34:01","slug":"diagonalize-a-symmetric-2x2-matrix-consider-a-2-x-2-matrix-a","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/05\/diagonalize-a-symmetric-2x2-matrix-consider-a-2-x-2-matrix-a\/","title":{"rendered":"Diagonalize a symmetric 2×2 matrix Consider a 2 x 2 matrix A ="},"content":{"rendered":"\n

HW12.4. Diagonalize a symmetric 2×2 matrix Consider a 2 x 2 matrix A = . Find an orthogonal matrix 2 x 2-matrix Q and a diagonal 2 x 2- matrix D such that A = QDQT. Q = D= Note: In order to be accepted as correct, all entries of the matrices A – QDQT and QTQ – I must have absolute value smaller than 0.05. Save & Grade 5 tries left Save only<\/p>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

To diagonalize a symmetric (2 \\times 2) matrix ( A ), we follow these steps:<\/p>\n\n\n\n

    \n
  1. Find the Eigenvalues<\/strong>: Solve ( \\det(A – \\lambda I) = 0 ) to find the eigenvalues ( \\lambda_1 ) and ( \\lambda_2 ).<\/li>\n\n\n\n
  2. Find the Eigenvectors<\/strong>: Solve ( (A – \\lambda I)x = 0 ) for each eigenvalue to get eigenvectors.<\/li>\n\n\n\n
  3. Construct the Orthogonal Matrix ( Q )<\/strong>: Normalize the eigenvectors and arrange them as columns in ( Q ).<\/li>\n\n\n\n
  4. Construct the Diagonal Matrix ( D )<\/strong>: Place the eigenvalues as diagonal entries.<\/li>\n\n\n\n
  5. Verify ( A = QD Q^T )<\/strong>: Check the reconstruction with a tolerance of 0.05.<\/li>\n\n\n\n
  6. Ensure ( Q ) is Orthogonal<\/strong>: Verify ( Q^T Q = I ).<\/li>\n<\/ol>\n\n\n\n

    Now, for the explanation:<\/p>\n\n\n\n

    \n\n\n\n

    Explanation:<\/h3>\n\n\n\n

    A symmetric matrix is always diagonalizable with an orthogonal matrix of eigenvectors. The process starts by computing the characteristic equation ( \\det(A – \\lambda I) = 0 ), yielding two real eigenvalues. Next, we determine corresponding eigenvectors and normalize them to form an orthogonal matrix ( Q ). The diagonal matrix ( D ) contains the eigenvalues. The relationship ( A = QD Q^T ) ensures the correctness of our diagonalization.<\/p>\n\n\n\n

    Orthogonal matrices have the property ( Q^T Q = I ), meaning they preserve angles and lengths. This is crucial in physics and engineering applications, such as quantum mechanics and principal component analysis.<\/p>\n\n\n\n

    To validate correctness, we compute ( A – QD Q^T ) and ( Q^T Q – I ) and ensure all entries are within a small tolerance (0.05). This confirms numerical accuracy.<\/p>\n\n\n\n

    Diagonalization simplifies matrix operations, making exponentiation and function computation more efficient. For example, powers of ( A ) can be computed as ( A^n = Q D^n Q^T ), where ( D^n ) is easy to compute. This is particularly useful in solving differential equations and modeling dynamical systems.<\/p>\n\n\n\n

    Thus, diagonalizing symmetric matrices is fundamental in linear algebra and has widespread applications in applied mathematics, physics, and computer science.<\/p>\n","protected":false},"excerpt":{"rendered":"

    HW12.4. Diagonalize a symmetric 2×2 matrix Consider a 2 x 2 matrix A = . Find an orthogonal matrix 2 x 2-matrix Q and a diagonal 2 x 2- matrix D such that A = QDQT. Q = D= Note: In order to be accepted as correct, all entries of the matrices A – QDQT […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","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":"","ast-disable-related-posts":"","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":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-187636","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/187636","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=187636"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/187636\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=187636"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=187636"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=187636"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}