{"id":3419,"date":"2015-04-12T01:22:08","date_gmt":"2015-04-12T06:22:08","guid":{"rendered":"http:\/\/www.ssc.wisc.edu\/~jfrees\/?page_id=3419"},"modified":"2015-04-19T14:31:38","modified_gmt":"2015-04-19T19:31:38","slug":"some-special-matrices","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/regression\/basic-linear-regression\/2-9-technical-supplement-elements-of-matrix-algebra\/some-special-matrices\/","title":{"rendered":"Some Special Matrices"},"content":{"rendered":"<ol>\n<li> A <em>square matrix<\/em> is a matrix where the number of rows equals the number of columns, that is, \\(n=k\\).\n <\/li>\n<li> The <em>diagonal numbers<\/em> of a square matrix are the numbers of a matrix where the row number equals the column number, for example, \\(a_{11}\\), \\(a_{22}\\), and so on. A <em>diagonal matrix<\/em> is a square matrix where all non-diagonal numbers are equal to 0. For example, \\begin{equation*} \\mathbf{A}=\\left( \\begin{array}{ccc} -1 &#038; 0 &#038; 0 \\\\ 0 &#038; 2 &#038; 0 \\\\ 0 &#038; 0 &#038; 3 \\end{array} \\right) . \\end{equation*}\n <\/li>\n<li> An <em>identity matrix<\/em> is a diagonal matrix where all the diagonal numbers are equal to 1. This special matrix is often denoted by \\(\\mathbf{I}\\).\n <\/li>\n<li> A<em> symmetric matrix<\/em> is a square matrix \\(\\mathbf{A}\\) such that the matrix remains unchanged if we interchange the roles of the rows and columns. More formally, a matrix \\(\\mathbf{A}\\) is symmetric if \\(\\mathbf{A=A} ^{\\prime }\\). For example, \\begin{equation*} \\mathbf{A}=\\left( \\begin{array}{ccc} 1 &#038; 2 &#038; 3 \\\\ 2 &#038; 4 &#038; 5 \\\\ 3 &#038; 5 &#038; 10 \\end{array} \\right) \\mathbf{=A}^{\\prime }. \\end{equation*} Note that a diagonal matrix is a symmetric matrix. <\/li>\n<\/ol>\n<p><div class=\"alignleft\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/regression\/basic-linear-regression\/2-9-technical-supplement-elements-of-matrix-algebra\/basic-definitions\/\" title=\"Basic Definitions\">&#9668 Previous page<\/a><\/div><div class=\"alignright\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/regression\/basic-linear-regression\/2-9-technical-supplement-elements-of-matrix-algebra\/basic-operations\/\" title=\"Basic Operations\">Next page &#9658<\/a><\/div><\/p>\n","protected":false},"excerpt":{"rendered":"<p>A square matrix is a matrix where the number of rows equals the number of columns, that is, \\(n=k\\). The diagonal numbers of a square matrix are the numbers of a matrix where the row &hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":3415,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-T9","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/3419"}],"collection":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/comments?post=3419"}],"version-history":[{"count":3,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/3419\/revisions"}],"predecessor-version":[{"id":3422,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/3419\/revisions\/3422"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/3415"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=3419"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}