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<div class="title">MATRIXPOWER Matrix Power Operator </div> </div>
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<div class="textblock"><p>Section: <a class="el" href="sec_operators.html">Mathematical Operators</a> </p>
<h1><a class="anchor" id="Usage"></a>
Usage</h1>
<p>The power operator for scalars and square matrices. This operator is really a combination of two operators, both of which have the same general syntax: </p>
<pre class="fragment"> y = a ^ b
</pre><p> The exact action taken by this operator, and the size and type of the output, depends on which of the two configurations of <code>a</code> and <code>b</code> is present: </p>
<ol>
<li>
<code>a</code> is a scalar, <code>b</code> is a square matrix </li>
<li>
<code>a</code> is a square matrix, <code>b</code> is a scalar </li>
</ol>
<h1><a class="anchor" id="Function"></a>
Internals</h1>
<p>In the first case that <code>a</code> is a scalar, and <code>b</code> is a square matrix, the matrix power is defined in terms of the eigenvalue decomposition of <code>b</code>. Let <code>b</code> have the following eigen-decomposition (problems arise with non-symmetric matrices <code>b</code>, so let us assume that <code>b</code> is symmetric): </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ b = E \begin{bmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \hdots & 0 & \lambda_n \end{bmatrix} E^{-1} \]" src="form_122.png"/>
</p>
<p> Then <code>a</code> raised to the power <code>b</code> is defined as </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ a^{b} = E \begin{bmatrix} a^{\lambda_1} & 0 & \cdots & 0 \\ 0 & a^{\lambda_2} & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \hdots & 0 & a^{\lambda_n} \end{bmatrix} E^{-1} \]" src="form_123.png"/>
</p>
<p> Similarly, if <code>a</code> is a square matrix, then <code>a</code> has the following eigen-decomposition (again, suppose <code>a</code> is symmetric): </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ a = E \begin{bmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \hdots & 0 & \lambda_n \end{bmatrix} E^{-1} \]" src="form_124.png"/>
</p>
<p> Then <code>a</code> raised to the power <code>b</code> is defined as </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ a^{b} = E \begin{bmatrix} \lambda_1^b & 0 & \cdots & 0 \\ 0 & \lambda_2^b & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \hdots & 0 & \lambda_n^b \end{bmatrix} E^{-1} \]" src="form_125.png"/>
</p>
<h1><a class="anchor" id="Examples"></a>
Examples</h1>
<p>We first define a simple <code>2 x 2</code> symmetric matrix</p>
<pre class="fragment">--> A = 1.5
A =
1.5000
--> B = [1,.2;.2,1]
B =
1.0000 0.2000
0.2000 1.0000
</pre><p>First, we raise <code>B</code> to the (scalar power) <code>A</code>:</p>
<pre class="fragment">--> C = B^A
C =
1.0150 0.2995
0.2995 1.0150
</pre><p>Next, we raise <code>A</code> to the matrix power <code>B</code>:</p>
<pre class="fragment">--> C = A^B
C =
1.5049 0.1218
0.1218 1.5049
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