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<span class="type">Section</span> <span class="codenumber">5.4</span> <span class="title">The number of eigenvalues of a matrix of order \(n\)</span>
</h2>
<p id="p-1145">If \(A\) is a matrix of order \(n\text{,}\) then \(p(\lambda)=\det(A-\lambda I)\) is a polynomial of degree \(n\text{.}\) Since a polynomial of degree \(n\) has at most \(n\) roots, the matrix has as most \(n\) eigenvalues.</p>
<article class="theorem theorem-like" id="NumberOfEigenvalues"><h6 class="heading">
<span class="type">Theorem</span><span class="space"> </span><span class="codenumber">5.4.1</span><span class="period">.</span><span class="space"> </span><span class="title">The number of eigenvalues of \(A\).</span>
</h6>
<p id="p-1146">A square matrix of order \(n\) has at most \(n\) eigenvalues.</p></article><article class="hiddenproof" id="proof-95"><a data-knowl="" class="id-ref proof-knowl original" data-refid="hk-proof-95"><h6 class="heading"><span class="type">Proof<span class="period">.</span></span></h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-proof-95"><article class="hiddenproof"><p id="p-1147">The characteristic polynomial \(p(\lambda)\) is of degree \(n\text{,}\) and such a polynomial has at most \(n\) real roots.</p></article></div>
<p id="p-1148">If we once again look at the matrix from <a class="xref" data-knowl="./knowl/EigenvalueExample4.html" title="Example 5.1.6: Eigenvalues of \(\begin{bmatrix}
1 \amp 1 \amp 1 \amp 1\\
0 \amp 2 \amp 2 \amp 2\\
0 \amp 0 \amp 3 \amp 3\\
0 \amp 0 \amp 0 \amp 4
\end{bmatrix}\)">Example 5.1.6</a>. we see that it is a matrix of order \(4\) with \(4\) eigenvalues given. <a class="xref" data-knowl="./knowl/NumberOfEigenvalues.html" title="Theorem 5.4.1: The number of eigenvalues of \(A\)">Theorem 5.4.1</a> tells us that there are no others.</p></section></div></main>
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