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<p>
Next: <a href="Matrix-Algebra.html" accesskey="n" rel="next">Linear Algebra with Diagonal/Permutation Matrices</a>, Up: <a href="Diagonal-and-Permutation-Matrices.html" accesskey="u" rel="up">Diagonal and Permutation Matrices</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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<hr>
<h3 class="section" id="Creating-and-Manipulating-Diagonal_002fPermutation-Matrices"><span>21.1 Creating and Manipulating Diagonal/Permutation Matrices<a class="copiable-link" href="#Creating-and-Manipulating-Diagonal_002fPermutation-Matrices"> ¶</a></span></h3>
<p>A diagonal matrix is defined as a matrix that has zero entries outside the main
diagonal; that is,
<code class="code">D(i,j) == 0</code> if <code class="code">i != j</code>.
Most often, square diagonal matrices are considered; however, the definition
can equally be applied to non-square matrices, in which case we usually speak
of a rectangular diagonal matrix.
</p>
<p>A permutation matrix is defined as a square matrix that has a single element
equal to unity in each row and each column; all other elements are zero. That
is, there exists a permutation (vector)
<code class="code">p</code> such that <code class="code">P(i,j) == 1</code> if <code class="code">j == p(i)</code> and
<code class="code">P(i,j) == 0</code> otherwise.
</p>
<p>Octave provides special treatment of real and complex rectangular diagonal
matrices, as well as permutation matrices. They are stored as special objects,
using efficient storage and algorithms, facilitating writing both readable and
efficient matrix algebra expressions in the Octave language. The special
treatment may be disabled by using the functions <em class="dfn">optimize_diagonal_matrix</em>
and <em class="dfn">optimize_permutation_matrix</em>.
</p>
<a class="anchor" id="XREFoptimize_005fdiagonal_005fmatrix"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-optimize_005fdiagonal_005fmatrix"><span><code class="def-type"><var class="var">val</var> =</code> <strong class="def-name">optimize_diagonal_matrix</strong> <code class="def-code-arguments">()</code><a class="copiable-link" href="#index-optimize_005fdiagonal_005fmatrix"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-optimize_005fdiagonal_005fmatrix-1"><span><code class="def-type"><var class="var">old_val</var> =</code> <strong class="def-name">optimize_diagonal_matrix</strong> <code class="def-code-arguments">(<var class="var">new_val</var>)</code><a class="copiable-link" href="#index-optimize_005fdiagonal_005fmatrix-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-optimize_005fdiagonal_005fmatrix-2"><span><code class="def-type"><var class="var">old_val</var> =</code> <strong class="def-name">optimize_diagonal_matrix</strong> <code class="def-code-arguments">(<var class="var">new_val</var>, "local")</code><a class="copiable-link" href="#index-optimize_005fdiagonal_005fmatrix-2"> ¶</a></span></dt>
<dd><p>Query or set whether a special space-efficient format is used for storing
diagonal matrices.
</p>
<p>The default value is true. If this option is set to false, Octave will store
diagonal matrices as full matrices.
</p>
<p>When called from inside a function with the <code class="code">"local"</code> option, the setting
is changed locally for the function and any subroutines it calls. The original
setting is restored when exiting the function.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="Ranges.html#XREFoptimize_005frange">optimize_range</a>, <a class="ref" href="#XREFoptimize_005fpermutation_005fmatrix">optimize_permutation_matrix</a>.
</p></dd></dl>
<a class="anchor" id="XREFoptimize_005fpermutation_005fmatrix"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-optimize_005fpermutation_005fmatrix"><span><code class="def-type"><var class="var">val</var> =</code> <strong class="def-name">optimize_permutation_matrix</strong> <code class="def-code-arguments">()</code><a class="copiable-link" href="#index-optimize_005fpermutation_005fmatrix"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-optimize_005fpermutation_005fmatrix-1"><span><code class="def-type"><var class="var">old_val</var> =</code> <strong class="def-name">optimize_permutation_matrix</strong> <code class="def-code-arguments">(<var class="var">new_val</var>)</code><a class="copiable-link" href="#index-optimize_005fpermutation_005fmatrix-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-optimize_005fpermutation_005fmatrix-2"><span><code class="def-type"><var class="var">old_val</var> =</code> <strong class="def-name">optimize_permutation_matrix</strong> <code class="def-code-arguments">(<var class="var">new_val</var>, "local")</code><a class="copiable-link" href="#index-optimize_005fpermutation_005fmatrix-2"> ¶</a></span></dt>
<dd><p>Query or set whether a special space-efficient format is used for storing
permutation matrices.
</p>
<p>The default value is true. If this option is set to false, Octave will store
permutation matrices as full matrices.
</p>
<p>When called from inside a function with the <code class="code">"local"</code> option, the setting
is changed locally for the function and any subroutines it calls. The original
setting is restored when exiting the function.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="Ranges.html#XREFoptimize_005frange">optimize_range</a>, <a class="ref" href="#XREFoptimize_005fdiagonal_005fmatrix">optimize_diagonal_matrix</a>.
</p></dd></dl>
<p>The space savings are significant as demonstrated by the following code.
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">x = diag (rand (10, 1));
xf = full (x);
sizeof (x)
⇒ 80
sizeof (xf)
⇒ 800
</pre></div></div>
<ul class="mini-toc">
<li><a href="Creating-Diagonal-Matrices.html" accesskey="1">Creating Diagonal Matrices</a></li>
<li><a href="Creating-Permutation-Matrices.html" accesskey="2">Creating Permutation Matrices</a></li>
<li><a href="Explicit-and-Implicit-Conversions.html" accesskey="3">Explicit and Implicit Conversions</a></li>
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