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<div class="title">PINV Moore-Penrose Pseudoinverse </div> </div>
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<div class="textblock"><p>Section: <a class="el" href="sec_array.html">Array Generation and Manipulations</a> </p>
<h1><a class="anchor" id="Usage"></a>
Usage</h1>
<p>Calculates the Moore-Penrose pseudoinverse of a matrix. The general syntax for its use is </p>
<pre class="fragment"> y = pinv(A,tol)
</pre><p> or for a default specification of the tolerance <code>tol</code>, </p>
<pre class="fragment"> y = pinv(A)
</pre><p> For any <code>m x n</code> matrix <code>A</code>, the Moore-Penrose pseudoinverse is the unique <code>n x m</code> matrix <code>B</code> that satisfies the following four conditions </p>
<ul>
<li>
<code>A B A = A</code> </li>
<li>
<code>B A B = B</code> </li>
<li>
<code>(A B)' = A B</code> </li>
<li>
<code>(B A)' = B A</code> </li>
</ul>
<p>Also, it is true that <code>B y</code> is the minimum norm, least squares solution to <code>A x = y</code>. The Moore-Penrose pseudoinverse is computed from the singular value decomposition of <code>A</code>, with singular values smaller than <code>tol</code> being treated as zeros. If <code>tol</code> is not specified then it is chosen as </p>
<pre class="fragment"> tol = max(size(A)) * norm(A) * teps(A).
</pre> <h1><a class="anchor" id="Function"></a>
Internals</h1>
<p>The calculation of the MP pseudo-inverse is almost trivial once the svd of the matrix is available. First, for a real, diagonal matrix with positive entries, the pseudo-inverse is simply </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \left(\Sigma^{+}\right)_{ii} = \begin{cases} 1/\sigma_{ii} & \sigma_{ii} > 0 \\ 0 & \mathrm{else} \end{cases} \]" src="form_5.png"/>
</p>
<p> One can quickly verify that this choice of matrix satisfies the four properties of the pseudoinverse. Then, the pseudoinverse of a general matrix <code>A = U S V'</code> is defined as </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ A^{+} = V S^{+} U' \]" src="form_6.png"/>
</p>
<p> and again, using the facts that <code>U' U = I</code> and <code>V V' = I</code>, one can quickly verify that this choice of pseudoinverse satisfies the four defining properties of the MP pseudoinverse. Note that in practice, the diagonal pseudoinverse <code>S^{+}</code> is computed with a threshold (the <code>tol</code> argument to <code>pinv</code>) so that singular values smaller than <code>tol</code> are treated like zeros. </p>
<h1><a class="anchor" id="Examples"></a>
Examples</h1>
<p>Consider a simple <code>1 x 2</code> matrix example, and note the various Moore-Penrose conditions:</p>
<pre class="fragment">--> A = float(rand(1,2))
A =
0.4840 0.0187
--> B = pinv(A)
B =
2.0630
0.0796
--> A*B*A
ans =
0.4840 0.0187
--> B*A*B
ans =
2.0630
0.0796
--> A*B
ans =
1.0000
--> B*A
ans =
0.9985 0.0385
0.0385 0.0015
</pre><p>To demonstrate that <code>pinv</code> returns the least squares solution, consider the following very simple case</p>
<pre class="fragment">--> A = float([1;1;1;1])
A =
1
1
1
1
</pre><p>The least squares solution to <code>A x = b</code> is just <code>x = mean(b)</code>, and computing the <code>pinv</code> of <code>A</code> demonstrates this</p>
<pre class="fragment">--> pinv(A)
ans =
0.2500 0.2500 0.2500 0.2500
</pre><p>Similarly, we can demonstrate the minimum norm solution with the following simple case</p>
<pre class="fragment">--> A = float([1,1])
A =
1 1
</pre><p>The solutions of <code>A x = 5</code> are those <code>x_1</code> and <code>x_2</code> such that <code>x_1 + x_2 = 5</code>. The norm of <code>x</code> is <code>x_1^ + x_2^2</code>, which is <code>x_1^2 + (5-x_1)^2</code>, which is minimized for <code>x_1 = x_2 = 2.5</code>:</p>
<pre class="fragment">--> pinv(A) * 5.0
ans =
2.5000
2.5000
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