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      <text>22.2.1.3 Matrix powers</text>
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      <text><![CDATA[<h4>Matrix powers</h4>
<p>If <span class="math inline">\(P^{-1}AP=D\)</span> then we can obtain <span class="math inline">\(A\)</span> (i.e. make <span class="math inline">\(A\)</span> the subject of this matrix equation) as follows:</p>

<p>Multiplying on the left by <span class="math inline">\(P\)</span> and on the right by <span class="math inline">\(P^{-1}\)</span> we obtain</p>
\[PP^{-1}APP^{-1}=PDP^{-1}\]
<p>Now using the fact that  <span class="math inline">\(PP^{-1}=P^{-1}P=I\)</span> we obtain</p>
\[IAI=PDP^{-1}\]
<p>and so</p>
\[A=PDP^{-1}\]

<p>We can use this result to obtain the <span><strong>powers</strong></span> of a square matrix, a process which is sometimes useful in control theory. Note that</p>
<p style="text-align: center;">\(A^2 = A \cdot A\), &emsp; \(A^3=A \cdot A \cdot A\) &emsp; etc.</p>
<p>Clearly, obtaining high powers of <span class="math inline">\(A\)</span> directly would in general involve many multiplications. The process is quite straightforward, however, for a diagonal matrix <span class="math inline">\(D\)</span>, as this next Exercise shows.</p>]]></text>
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