{"id":4994,"date":"2021-09-04T00:16:42","date_gmt":"2021-09-03T18:46:42","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4994"},"modified":"2021-11-26T20:44:38","modified_gmt":"2021-11-26T15:14:38","slug":"symmetric-and-skew-symmetric-matrices","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/symmetric-and-skew-symmetric-matrices\/","title":{"rendered":"Symmetric and Skew Symmetric Matrices"},"content":{"rendered":"

Here you will learn what are symmetric and skew symmetric matrices with examples.<\/p>\n

Let’s begin –<\/p>\n

Symmetric and Skew Symmetric Matrices<\/h2>\n

Symmetric Matrix<\/strong><\/h4>\n

A square matrix A = \\([a_{ij}]\\) is called a symmetric matrix, if<\/p>\n

\n

\\(A^T\\) = A or \\(a_{ij}\\) = \\(a_{ji}\\) f<\/a>or all i, j.\u00a0<\/p>\n<\/blockquote>\n

for example<\/strong>, the matrix A = \\(\\begin{bmatrix} 3 & -1 & 1 \\\\ -1 &\u00a0 2 & 5 \\\\ 1 & 5 &\u00a0 -2\u00a0 \\end{bmatrix}\\) is symmetric, because<\/p>\n

\\(a_{12}\\) = -1 = \\(a_{21}\\), \\(a_{13}\\) = 1 = \\(a_{31}\\), \\(a_{23}\\) = 5 = \\(a_{32}\\) i.e. \\(a_{ij}\\) = \\(a_{ji}\\) for all i, j.<\/p>\n

It follows from the definition of a symmetric matrix that A is symmetric, iff<\/p>\n

\\(a_{ij}\\) = \\(a_{ji}\\)\u00a0 \u00a0for all i, j.<\/p>\n

\\(\\iff\\)\u00a0 \\(A_{ij}\\) = \\((A^T)_{ij}\\)\u00a0 for all i, j<\/p>\n

\\(\\iff\\)\u00a0 A = \\(A^T\\)<\/p>\n

Thus, a square matrix A is a symmetric matrix iff \\(A^T\\) = A.<\/p>\n

Matrices A = \\(\\begin{bmatrix} a & h & g \\\\ h &\u00a0 b & f \\\\ g & f &\u00a0 c\u00a0 \\end{bmatrix}\\), B = \\(\\begin{bmatrix} 2 + i & 1 & 3 \\\\ 1 &\u00a0 2 & 3 + 2i \\\\ 3 & 3 + 2i &\u00a0 4\u00a0 \\end{bmatrix}\\) are symmetric matrices because \\(A^T\\) = A and \\(B^T\\) = B.<\/p>\n

Skew-Symmetric Matrix<\/strong><\/h4>\n

A square matrix A = \\([a_{ij}]\\) is called a skew-symmetric matrix, if<\/p>\n

\n

\\(A^T\\) = -A or \\(a_{ij}\\) = -\\(a_{ji}\\) for all i, j.\u00a0<\/p>\n<\/blockquote>\n

for example<\/strong>, the matrix A = \\(\\begin{bmatrix} 0 & 2 & -3 \\\\ -2 &\u00a0 0 & 5 \\\\ 3 & -5 &\u00a0 0\u00a0 \\end{bmatrix}\\) is skew symmetric, because<\/p>\n

\\(a_{12}\\) = 2, \\(a_{21}\\) = -2 \\(\\implies\\) \\(a_{12}\\) = -\\(a_{21}\\) \\(a_{13}\\) = -3, \\(a_{31}\\) = 3 \\(\\implies\\) \\(a_{13}\\) = -\\(a_{31}\\)\u00a0<\/p>\n

and,\u00a0 \\(a_{23}\\) = 5,\u00a0 \\(a_{32}\\) = – 5 \\(\\implies\\) \\(a_{23}\\) = -\\(a_{32}\\)<\/p>\n

It follows from the definition of a skew symmetric matrix that A is skew symmetric, iff<\/p>\n

\\(a_{ij}\\) = -\\(a_{ji}\\)\u00a0 \u00a0for all i, j.<\/p>\n

\\(\\iff\\)\u00a0 \\(A_{ij}\\) = -\\((A^T)_{ij}\\)\u00a0 for all i, j<\/p>\n

\\(\\iff\\)\u00a0 A = -\\(A^T\\)<\/p>\n

Thus, a square matrix A is a skew symmetric matrix iff \\(A^T\\) = -A.<\/p>\n

Matrices A = \\(\\begin{bmatrix} 0 & 2i & 3 \\\\ -2i &\u00a0 0 & 4 \\\\ -3 & -4 & 0\u00a0 \\end{bmatrix}\\), B = \\(\\begin{bmatrix} 0 & -3 & 5 \\\\ 3 &\u00a0 0 & 2 \\\\ -5 & -2 &\u00a0 0\u00a0 \\end{bmatrix}\\) are skew symmetric matrices because \\(A^T\\) = -A and \\(B^T\\) = -B.<\/p>\n\n\n

\n
Next – How to Find Trace of Matrix \u2013 Properties & Example<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Transpose of Matrix – Definition & Example<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what are symmetric and skew symmetric matrices with examples. Let’s begin – Symmetric and Skew Symmetric Matrices Symmetric Matrix A square matrix A = \\([a_{ij}]\\) is called a symmetric matrix, if \\(A^T\\) = A or \\(a_{ij}\\) = \\(a_{ji}\\) for all i, j.\u00a0 for example, the matrix A = \\(\\begin{bmatrix} 3 & …<\/p>\n

Symmetric and Skew Symmetric Matrices<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[34],"tags":[481,479,480],"yoast_head":"\nSymmetric and Skew Symmetric Matrices - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what are symmetric and skew symmetric matrices with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/symmetric-and-skew-symmetric-matrices\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Symmetric and Skew Symmetric Matrices - 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