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<div class="title">SVD Singular Value Decomposition of a Matrix </div> </div>
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<div class="textblock"><p>Section: <a class="el" href="sec_transforms.html">Transforms/Decompositions</a> </p>
<h1><a class="anchor" id="Usage"></a>
Usage</h1>
<p>Computes the singular value decomposition (SVD) of a matrix. The <code>svd</code> function has three forms. The first returns only the singular values of the matrix: </p>
<pre class="fragment"> s = svd(A)
</pre><p> The second form returns both the singular values in a diagonal matrix <code>S</code>, as well as the left and right eigenvectors. </p>
<pre class="fragment"> [U,S,V] = svd(A)
</pre><p> The third form returns a more compact decomposition, with the left and right singular vectors corresponding to zero singular values being eliminated. The syntax is </p>
<pre class="fragment"> [U,S,V] = svd(A,0)
</pre> <h1><a class="anchor" id="Function"></a>
Internals</h1>
<p>Recall that <code>sigma_i</code> is a singular value of an <code>M x N</code> matrix <code>A</code> if there exists two vectors <code>u_i, v_i</code> where <code>u_i</code> is of length <code>M</code>, and <code>v_i</code> is of length <code>u_i</code> and </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ A v_i = \sigma_i u_i \]" src="form_161.png"/>
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<p> and generally </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ A = \sum_{i=1}^{K} \sigma_i u_i*v_i', \]" src="form_162.png"/>
</p>
<p> where <code>K</code> is the rank of <code>A</code>. In matrix form, the left singular vectors <code>u_i</code> are stored in the matrix <code>U</code> as </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ U = [u_1,\ldots,u_m], V = [v_1,\ldots,v_n] \]" src="form_163.png"/>
</p>
<p> The matrix <code>S</code> is then of size <code>M x N</code> with the singular values along the diagonal. The SVD is computed using the <code>LAPACK</code> class of functions <code>GESVD</code> (Note that this has changed. Previous versions of FreeMat used <code>GESDD</code>, which yields a valid, but slightly different choice of the decomposition. Starting in version 4, it was changed to <code>GESVD</code> to improve compatibility. </p>
<h1><a class="anchor" id="Examples"></a>
Examples</h1>
<p>Here is an example of a partial and complete singular value decomposition.</p>
<pre class="fragment">--> A = float(randn(2,3))
A =
0.1962 -1.7828 -1.0621
-0.6022 -0.6335 0.5810
--> [U,S,V] = svd(A)
U =
-0.9929 -0.1189
-0.1189 0.9929
S =
2.0957 0 0
0 1.0268 0
V =
-0.0588 -0.6051 0.7940
0.8806 -0.4061 -0.2443
0.4702 0.6848 0.5567
--> U*S*V'
ans =
0.1962 -1.7828 -1.0621
-0.6022 -0.6335 0.5810
--> svd(A)
ans =
2.0957
1.0268
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