https://artofproblemsolving.com/wiki/index.php?title=Identity_matrix&feed=atom&action=history Identity matrix - Revision history 2022-01-24T22:37:57Z Revision history for this page on the wiki MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=Identity_matrix&diff=33898&oldid=prev Azjps: create stub 2010-03-17T04:00:42Z <p>create stub</p> <p><b>New page</b></p><div>In [[linear algebra]], the square '''identity matrix''' is a &lt;math&gt;n \times n&lt;/math&gt; matrix with &lt;math&gt;1&lt;/math&gt;s in its [[main diagonal]] and &lt;math&gt;0&lt;/math&gt;s in every other entry. It is usually denoted &lt;math&gt;I_n&lt;/math&gt;. <br /> <br /> &lt;cmath&gt;I_n = \begin{pmatrix}1 &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; 1 &amp; 0 &amp; \cdots &amp; 0\\ 0 &amp; 0 &amp; 1 &amp; \cdots &amp; 0 \\ \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\ 0 &amp; 0 &amp; 0 &amp; \cdots &amp; 1\end{pmatrix}.&lt;/cmath&gt;<br /> <br /> The corresponding linear map is the identity map. For any &lt;math&gt;n \times n&lt;/math&gt; matrix &lt;math&gt;A&lt;/math&gt;, we have &lt;math&gt;AI_n = I_nA = A&lt;/math&gt;. The [[matrix inverse|inverse]] &lt;math&gt;A^{-1}&lt;/math&gt; of &lt;math&gt;A&lt;/math&gt; is the unique matrix such that &lt;math&gt;AA^{-1} = A^{-1}A = I_n&lt;/math&gt;. <br /> <br /> The [[determinant]] of &lt;math&gt;I_n&lt;/math&gt; is &lt;math&gt;1&lt;/math&gt;. &lt;math&gt;I_n&lt;/math&gt; has only one [[eigenvalue]] &lt;math&gt;1&lt;/math&gt;, occurring with multiplicity &lt;math&gt;n&lt;/math&gt;. Hence, any &lt;math&gt;n \times n&lt;/math&gt; matrix is in the corresponding eigenspace. The [[characteristic polynomial]] of &lt;math&gt;I_n&lt;/math&gt; is &lt;math&gt;P_{I}(t) = (t-1)^n&lt;/math&gt;, and the [[minimal polynomial]] is &lt;math&gt;t - 1&lt;/math&gt;. <br /> <br /> [[Category:Linear algebra]]</div> Azjps