{"id":3417,"date":"2015-04-12T01:21:10","date_gmt":"2015-04-12T06:21:10","guid":{"rendered":"http:\/\/www.ssc.wisc.edu\/~jfrees\/?page_id=3417"},"modified":"2015-04-19T14:31:34","modified_gmt":"2015-04-19T19:31:34","slug":"basic-definitions","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/regression\/basic-linear-regression\/2-9-technical-supplement-elements-of-matrix-algebra\/basic-definitions\/","title":{"rendered":"Basic Definitions"},"content":{"rendered":"<p>A <em>matrix<\/em> is a rectangular array of numbers arranged in rows and columns (the plural of matrix is matrices). For example, consider the income and age of 3 people. \\begin{equation*} \\mathbf{A}= \\begin{array}{c} Row~1 \\\\ Row~2 \\\\ Row~3 \\end{array} \\overset{ \\begin{array}{cc} ~~~Col~1~ &#038; Col~2 \\end{array} }{\\left( \\begin{array}{cc} 6,000 &#038; 23 \\\\ 13,000 &#038; 47 \\\\ 11,000 &#038; 35 \\end{array} \\right) } \\end{equation*} <\/p>\n<p>  Here, column 1 represents income and column 2 represents age. Each row corresponds to an individual. For example, the first individual is 23 years old with an income of $6,000. <\/p>\n<p> The number of rows and columns is called the <em>dimension<\/em> of the matrix. For example, the dimension of the matrix \\(\\mathbf{A}\\) above is \\(3\\times 2\\) (read 3 &#8220;by&#8221; 2). This stands for 3 rows and 2 columns. If we were to represent the income and age of 100 people, then the dimension of the matrix would be \\(100\\times 2\\). <\/p>\n<p> It is convenient to represent a matrix using the notation \\begin{equation*} \\mathbf{A}=\\left( \\begin{array}{cc} a_{11} &#038; a_{12} \\\\ a_{21} &#038; a_{22} \\\\ a_{31} &#038; a_{31} \\end{array} \\right) . \\end{equation*} Here, \\(a_{ij}\\) is the symbol for the number in the \\(i\\)th row and \\(j\\)th column of \\(\\mathbf{A}\\). In general, we work with matrices of the form \\begin{equation*} \\mathbf{A}=\\left( \\begin{array}{cccc} a_{11} &#038; a_{12} &#038; \\cdots &#038; a_{1k} \\\\ \\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\ a_{n1} &#038; a_{n2} &#038; \\cdots &#038; a_{nk} \\end{array} \\right) . \\end{equation*} In this case, the matrix \\(\\mathbf{A}\\) has dimension \\(n\\times k\\). <\/p>\n<p> A <em>vector<\/em> is a special matrix. A <em>row vector<\/em> is a matrix containing only 1 row (\\(k=1\\)). A <em>column vector<\/em> is a matrix containing only 1 column (\\(n=1\\)). For example, \\begin{equation*} \\text{column vector}\\rightarrow \\left( \\begin{array}{c} 2 \\\\ 3 \\\\ 4 \\\\ 5 \\\\ 6 \\end{array} \\right) ~~~~\\text{row vector}\\rightarrow \\left( \\begin{array}{ccccc} 2 &#038; 3 &#038; 4 &#038; 5 &#038; 6 \\end{array} \\right) . \\end{equation*} Notice above that the row vector takes much less room on a printed page than the corresponding column vector. A basic operation that relates these two quantities is the <em>transpose<\/em>. The transpose of a matrix \\(\\mathbf{A}\\) is defined by interchanging the rows and columns and is denoted by \\(\\mathbf{A }^{\\prime }\\) (or \\(\\mathbf{A}^{T}\\)). For example, <\/p>\n<p> \\begin{equation*} \\mathbf{A}=\\left( \\begin{array}{cc} 6,000 &#038; 23 \\\\ 13,000 &#038; 47 \\\\ 11,000 &#038; 35 \\end{array} \\right) ~~~\\mathbf{A}^{\\prime }=\\left( \\begin{array}{ccc} 6,000 &#038; 13,000 &#038; 11,000 \\\\ 23 &#038; 47 &#038; 35 \\end{array} \\right) . \\end{equation*} Thus, if \\(\\mathbf{A}\\) has dimension \\(n\\times k\\), then \\(\\mathbf{A}^{\\prime }\\) has dimensions \\(k\\times n\\). <\/p>\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\/\" title=\"2.9 Technical Supplement &#8211; Elements of Matrix Algebra\">&#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\/some-special-matrices\/\" title=\"Some Special Matrices\">Next page &#9658<\/a><\/div><\/p>\n","protected":false},"excerpt":{"rendered":"<p>A matrix is a rectangular array of numbers arranged in rows and columns (the plural of matrix is matrices). For example, consider the income and age of 3 people. \\begin{equation*} \\mathbf{A}= \\begin{array}{c} Row~1 \\\\ Row~2 &hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":3415,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-T7","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/3417"}],"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=3417"}],"version-history":[{"count":1,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/3417\/revisions"}],"predecessor-version":[{"id":3418,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/3417\/revisions\/3418"}],"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=3417"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}