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      <text>22.1.4.1 Exercises</text>
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      <text><![CDATA[<p class="HELM_exercise">Paper Exercise</p>
<p>If <span class="math inline">\(\lambda_1,\lambda_2,\ldots \lambda_n\)</span> are the eigenvalues of a matrix <span class="math inline">\(A,\)</span>, get a piece of paper and prove the following:<br></p>
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<li><p><span class="math inline">\(A^T\)</span> has eigenvalues <span class="math inline">\(\lambda_1,\lambda_2,\ldots
\lambda_n\)</span>.</p></li>
<li><p>If <span class="math inline">\(A\)</span> is upper triangular, then its eigenvalues are exactly the main diagonal entries.</p></li>
<li><p>The inverse matrix <span class="math inline">\(A^{-1}\)</span> has eigenvalues <span class="math inline">\({{1\over
\lambda_1}},\ {{1\over \lambda_2}},\ldots {{1\over
\lambda_n}}\)</span>.</p></li>
<li><p>The matrix <span class="math inline">\(A-kI\)</span> has eigenvalues <span class="math inline">\(\lambda_1-k,\lambda_2-k,\ldots \lambda_n-k\)</span>.</p></li>
<li><p>(Harder) The matrix <span class="math inline">\(A^2\)</span> has eigenvalues <span class="math inline">\(\lambda_1^2,\lambda_2^2,\ldots \lambda_n^2\)</span>.</p></li>
<li><p>(Harder) The matrix <span class="math inline">\(A^k\)</span> (<span class="math inline">\(k\)</span> a non-negative integer) has eigenvalues <span class="math inline">\(\lambda_1^k,\lambda_2^k,\ldots \lambda_n^k.\)</span></p></li>
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<p>N.B. Some of these results are useful in the numerical calculation of eigenvalues.&nbsp;</p><p><b>We give the solution on the next page. Please try to complete the problem before referencing the solution.</b></p>]]></text>
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