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      <text>7.1.2.1 Definitions</text>
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      <text><![CDATA[<h3><span style="color: inherit; font-family: inherit;">Definitions</span><br></h3>
<p>An array of numbers, rectangular in shape, is called a&nbsp;<b>matrix</b>.&nbsp; The first matrix below has \(3\) rows and \(2\) columns and is said to be a '\(3\) by \(2\)' matrix (written \(3\times 2\)). The second matrix is a '\(2\) by \(4\)' matrix (written \(2\times 4\)).</p><p>\[\left[\begin{array}{rr}1&amp;4\\ -2&amp;3\\ 2&amp;1\end{array}\right]\ \qquad\qquad\left[\begin{array}{cccc}1&amp;2&amp;3&amp;4\\ 5&amp;6&amp;7&amp;9\end{array}\right]\]</p><p>The general \(3\times 3\) matrix can be written</p><p>\[A=\left[\begin{array}{lll}a_{11}&amp;a_{12}&amp;a_{13}\\ a_{21}&amp;a_{22}&amp;a_{23}\\a_{31}&amp;a_{32}&amp;a_{33}\end{array}\right]\]</p><p>where \(a_{ij}\) denotes the element in row \(i\), column \(j\).</p><p>For example in the matrix:</p><p>\[A=\left[\begin{array}{lll}\hfill 0&amp;\hfill -1&amp;\hfill -3\\ \hfill0&amp;\hfill 6&amp;\hfill -12\\ \hfill 5&amp;\hfill 7&amp;\hfill 123\end{array}\right]\]</p><p>\[a_{11}= 0,\qquad a_{12}=-1,\qquad a_{13}=-3,\quad\dots \quad a_{22}=6,\quad \dots \quad a_{32}=7,\quad a_{33}=123\]</p>
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<h4>Key Point</h4>
<h5 style="text-align:center;">The General Matrix</h5>
<p style="text-align: left;"><span></span></p><p style="">A general \(m\times n\) matrix \(A\)&nbsp; has \(m\) rows and \(n\) columns. The entries in the matrix \(A\) are called the <b>elements</b>&nbsp;of \(A\). In matrix \(A\) the element in row \(i\) and column \(j\) is denoted by \(a_{ij}\).</p><p></p>
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<p></p><p>A matrix with only one column is called a <b>column vector</b>&nbsp;(or <b>column matrix</b>). For example, \(\left[\begin{array}{c}x_1\\ x_2\\ x_3\end{array}\right]\) and \(\left[\begin{array}{c}3\\ 4\\ 5\end{array}\right]\) are both \(3\times 1\) column vectors. A matrix with only one row is called a <b>row vector</b>&nbsp;(or <b>row matrix</b>). For example,&nbsp; \([2, -3, 8, 9]\) is a \(1\times 4\) row vector. Often the entries in a row vector are separated by commas for clarity.</p><h4>Square matrices</h4><p></p><p>When the number of rows is the same as the number of columns, i.e. \(m=n\), the matrix is said to be <b>square</b>&nbsp;and of <b>order&nbsp;</b>\(n\) (or \(m\)).</p><p></p><ul><li>In an \(n\times n\) square matrix \(A\), the <b>leading diagonal</b>&nbsp;(or <b>principal diagonal</b>) is the `north-west to south-east' collection of elements \(a_{11},a_{22},\ldots , a_{nn}.\)&nbsp; The sum of the elements in the leading diagonal of \(A\) is called the <b>trace</b>&nbsp;of the matrix, denoted by \(\text{tr}(A)\).&nbsp;</li></ul><p></p><p>\[A=\left[\begin{matrix} a_{11}&amp;a_{12}&amp;\dots&amp;a_{1n}\\a_{21}&amp;a_{22}&amp;\dots&amp;a_{2n}\\\vdots&amp;\vdots&amp;\vdots&amp;\vdots\\a_{n1}&amp;a_{n2}&amp;\dots&amp;a_{nn}\end{matrix}\right] \qquad\qquad \text{tr}(A)=a_{11}+a_{22}+\dots +a_{nn}\]</p><p></p><p></p><ul><li>A square matrix in which all the elements below the leading diagonal are zero is called an <b>upper triangular matrix</b>, often denoted by \(U\).</li>
</ul>\[U=\left[\begin{matrix}u_{11}&amp;u_{12}&amp;\dots&amp;\dots&amp;u_{1n}\\0&amp;u_{22}&amp;\dots&amp;\dots&amp;u_{2n}\\0&amp;0&amp;\dots&amp;\vdots&amp;\vdots\\0&amp;0&amp;\dots&amp;0&amp;u_{nn}\end{matrix}\right] \qquad\qquad u_{ij}=0\quad \text{when}\ \ i&gt;j\]<br><ul><li>A square matrix in which all the elements above the leading diagonal are zero is called a <b>lower triangular matrix</b>, often denoted by \(L\).</li><p></p></ul>\[L=\left[\begin{matrix}l_{11}&amp;0&amp;0&amp;\dots&amp;0\\l_{21}&amp;l_{22}&amp;0&amp;\dots&amp;0\\\vdots&amp;\vdots&amp;\dots&amp;\dots&amp;0\\l_{n1}&amp;l_{n2}&amp;\vdots&amp;\dots&amp;l_{nn}\end{matrix}\right] \qquad\qquad l_{ij}=0\quad \text{when}\ \ i&lt;j\]<p></p><ul><li>A square matrix where all the non-zero elements are along the leading diagonal is called a diagonal matrix, often denoted by \(D\).&nbsp;</li></ul><p></p><p>\[ D=\left[\begin{matrix}d_{11}&amp;0&amp;0&amp;\dots&amp;0\\0&amp;d_{22}&amp;0&amp;\dots&amp;0\\0&amp;0&amp;\dots&amp;\dots&amp;0\\0&amp;0&amp;0&amp;\dots&amp;d_{nn}\end{matrix}\right] \qquad\qquad d_{ij}=0\quad \text{when}\ \ i\ne j\]</p><p></p><h4>Some examples of matrices and their classification</h4><p>\(\qquad A=\left[\begin{array}{ccc} 1&amp;2&amp;3\\ 4&amp;5&amp;6\end{array}\right]\) is \(2\times 3\).&nbsp; It is not square.</p><p>\(\qquad B=\left[\begin{array}{cc}1&amp;2\\ 3&amp;4\end{array}\right]\) is \(2\times 2\). It is square.&nbsp;</p><p>Also, \(\text{tr}(A)\) does not exist as it is not a square matrix, but \(\text{tr}(B)=1+4=5\).</p><p>\(\qquad C=\left[\begin{array}{rrr}1&amp;2&amp;3\\ 0&amp;-2&amp;-5\\ 0&amp;0&amp;1\end{array}\right]\) and \(D=\left[\begin{array}{rrr} 4&amp;0&amp;3\\ 0&amp;-2&amp;5\\ 0&amp;0&amp;1\end{array}\right]\) are both \(3\times 3\), square and upper triangular.</p><p>Also, \(\text{tr}(C)=0\) and \(\text{tr}(D)=3\).</p><p>\(\qquad E=\left[\begin{array}{rrr}1&amp;0&amp;0\\ 2&amp;-2&amp;0\\ 3&amp;-5&amp;1\end{array}\right]\) and \(F=\left[\begin{array}{rrr}-1&amp;0&amp;0\\ 1&amp;4&amp;0\\ 0&amp;1&amp;1\end{array}\right]\) are both \(3\times 3\), square and lower triangular.</p><p>Also, \(\text{tr}(E)=0\) and \(\text{tr}(F)=4\).</p><p>\(\qquad G=\left[\begin{array}{rrr}1&amp;0&amp;0\\ 0&amp;2&amp;0\\ 0&amp;0&amp;-3\end{array}\right]\) and \(H=\left[\begin{array}{ccc}4&amp;0&amp;0\\ 0&amp;2&amp;0\\ 0&amp;0&amp;0\end{array}\right]\) are both \(3\times 3\), square and diagonal.</p><p>Also, \(\text{tr}(G)=0\) and \(\text{tr}(H)=6\).</p><ul><p></p></ul>]]></text>
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