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<a name="Basic-Matrix-Functions"></a>
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<p>
Next: <a href="Matrix-Factorizations.html#Matrix-Factorizations" accesskey="n" rel="next">Matrix Factorizations</a>, Previous: <a href="Techniques-Used-for-Linear-Algebra.html#Techniques-Used-for-Linear-Algebra" accesskey="p" rel="prev">Techniques Used for Linear Algebra</a>, Up: <a href="Linear-Algebra.html#Linear-Algebra" accesskey="u" rel="up">Linear Algebra</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Basic-Matrix-Functions-1"></a>
<h3 class="section">18.2 Basic Matrix Functions</h3>
<a name="index-matrix-functions_002c-basic"></a>

<a name="XREFbalance"></a><dl>
<dt><a name="index-balance"></a>: <em><var>AA</var> =</em> <strong>balance</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-balance-1"></a>: <em><var>AA</var> =</em> <strong>balance</strong> <em>(<var>A</var>, <var>opt</var>)</em></dt>
<dt><a name="index-balance-2"></a>: <em>[<var>DD</var>, <var>AA</var>] =</em> <strong>balance</strong> <em>(<var>A</var>, <var>opt</var>)</em></dt>
<dt><a name="index-balance-3"></a>: <em>[<var>D</var>, <var>P</var>, <var>AA</var>] =</em> <strong>balance</strong> <em>(<var>A</var>, <var>opt</var>)</em></dt>
<dt><a name="index-balance-4"></a>: <em>[<var>CC</var>, <var>DD</var>, <var>AA</var>, <var>BB</var>] =</em> <strong>balance</strong> <em>(<var>A</var>, <var>B</var>, <var>opt</var>)</em></dt>
<dd>
<p>Balance the matrix <var>A</var> to reduce numerical errors in future
calculations.
</p>
<p>Compute <code><var>AA</var> = <var>DD</var> \ <var>A</var> * <var>DD</var></code> in which <var>AA</var>
is a matrix whose row and column norms are roughly equal in magnitude, and
<code><var>DD</var> = <var>P</var> * <var>D</var></code>, in which <var>P</var> is a permutation
matrix and <var>D</var> is a diagonal matrix of powers of two.  This allows the
equilibration to be computed without round-off.  Results of eigenvalue
calculation are typically improved by balancing first.
</p>
<p>If two output values are requested, <code>balance</code> returns
the diagonal <var>D</var> and the permutation <var>P</var> separately as vectors.
In this case, <code><var>DD</var> = eye(n)(:,<var>P</var>) * diag (<var>D</var>)</code>, where
<em>n</em> is the matrix size.
</p>
<p>If four output values are requested, compute <code><var>AA</var> =
<var>CC</var>*<var>A</var>*<var>DD</var></code> and <code><var>BB</var> = <var>CC</var>*<var>B</var>*<var>DD</var></code>,
in which <var>AA</var> and <var>BB</var> have nonzero elements of approximately the
same magnitude and <var>CC</var> and <var>DD</var> are permuted diagonal matrices as
in <var>DD</var> for the algebraic eigenvalue problem.
</p>
<p>The eigenvalue balancing option <var>opt</var> may be one of:
</p>
<dl compact="compact">
<dt><code>&quot;noperm&quot;</code>, <code>&quot;S&quot;</code></dt>
<dd><p>Scale only; do not permute.
</p>
</dd>
<dt><code>&quot;noscal&quot;</code>, <code>&quot;P&quot;</code></dt>
<dd><p>Permute only; do not scale.
</p></dd>
</dl>

<p>Algebraic eigenvalue balancing uses standard <small>LAPACK</small> routines.
</p>
<p>Generalized eigenvalue problem balancing uses Ward&rsquo;s algorithm
(SIAM Journal on Scientific and Statistical Computing, 1981).
</p></dd></dl>


<a name="XREFbandwidth"></a><dl>
<dt><a name="index-bandwidth"></a>: <em><var>bw</var> =</em> <strong>bandwidth</strong> <em>(<var>A</var>, <var>type</var>)</em></dt>
<dt><a name="index-bandwidth-1"></a>: <em>[<var>lower</var>, <var>upper</var>] =</em> <strong>bandwidth</strong> <em>(<var>A</var>)</em></dt>
<dd><p>Compute the bandwidth of <var>A</var>.
</p>
<p>The <var>type</var> argument is the string <code>&quot;lower&quot;</code> for the lower
bandwidth and <code>&quot;upper&quot;</code> for the upper bandwidth.  If no <var>type</var> is
specified return both the lower and upper bandwidth of <var>A</var>.
</p>
<p>The lower/upper bandwidth of a matrix is the number of
subdiagonals/superdiagonals with nonzero entries.
</p>

<p><strong>See also:</strong> <a href="Predicates-for-Numeric-Objects.html#XREFisbanded">isbanded</a>, <a href="Predicates-for-Numeric-Objects.html#XREFisdiag">isdiag</a>, <a href="Predicates-for-Numeric-Objects.html#XREFistril">istril</a>, <a href="Predicates-for-Numeric-Objects.html#XREFistriu">istriu</a>.
</p></dd></dl>


<a name="XREFcond"></a><dl>
<dt><a name="index-cond"></a>: <em></em> <strong>cond</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-cond-1"></a>: <em></em> <strong>cond</strong> <em>(<var>A</var>, <var>p</var>)</em></dt>
<dd><p>Compute the <var>p</var>-norm condition number of a matrix with respect to
inversion.
</p>
<p><code>cond (<var>A</var>)</code> is defined as
<code>norm (<var>A</var>, <var>p</var>) * norm (inv (<var>A</var>), <var>p</var>)</code>.
</p>
<p>By default, <code><var>p</var> = 2</code> is used which implies a (relatively slow)
singular value decomposition.  Other possible selections are
<code><var>p</var> = 1, Inf, &quot;fro&quot;</code> which are generally faster.  See <code>norm</code>
for a full discussion of possible <var>p</var> values.
</p>
<p>The condition number of a matrix quantifies the sensitivity of the matrix
inversion operation when small changes are made to matrix elements.  Ideally
the condition number will be close to 1.  When the number is large this
indicates small changes (such as underflow or round-off error) will produce
large changes in the resulting output.  In such cases the solution results
from numerical computing are not likely to be accurate.
</p>
<p><strong>See also:</strong> <a href="Sparse-Linear-Algebra.html#XREFcondest">condest</a>, <a href="#XREFrcond">rcond</a>, <a href="#XREFcondeig">condeig</a>, <a href="#XREFnorm">norm</a>, <a href="Matrix-Factorizations.html#XREFsvd">svd</a>.
</p></dd></dl>


<a name="XREFcondeig"></a><dl>
<dt><a name="index-condeig"></a>: <em><var>c</var> =</em> <strong>condeig</strong> <em>(<var>a</var>)</em></dt>
<dt><a name="index-condeig-1"></a>: <em>[<var>v</var>, <var>lambda</var>, <var>c</var>] =</em> <strong>condeig</strong> <em>(<var>a</var>)</em></dt>
<dd><p>Compute condition numbers of a matrix with respect to eigenvalues.
</p>
<p>The condition numbers are the reciprocals of the cosines of the angles
between the left and right eigenvectors; Large values indicate that the
matrix has multiple distinct eigenvalues.
</p>
<p>The input <var>a</var> must be a square numeric matrix.
</p>
<p>The outputs are:
</p>
<ul>
<li> <var>c</var> is a vector of condition numbers for the eigenvalues of
<var>a</var>.

</li><li> <var>v</var> is the matrix of right eigenvectors of <var>a</var>.  The result is
equivalent to calling <code>[<var>v</var>, <var>lambda</var>] = eig (<var>a</var>)</code>.

</li><li> <var>lambda</var> is the diagonal matrix of eigenvalues of <var>a</var>.  The
result is equivalent to calling
<code>[<var>v</var>, <var>lambda</var>] = eig (<var>a</var>)</code>.
</li></ul>

<p>Example
</p>
<div class="example">
<pre class="example">a = [1, 2; 3, 4];
c = condeig (a)
&rArr; [1.0150; 1.0150]
</pre></div>

<p><strong>See also:</strong> <a href="#XREFeig">eig</a>, <a href="#XREFcond">cond</a>, <a href="#XREFbalance">balance</a>.
</p></dd></dl>


<a name="XREFdet"></a><dl>
<dt><a name="index-det"></a>: <em></em> <strong>det</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-det-1"></a>: <em>[<var>d</var>, <var>rcond</var>] =</em> <strong>det</strong> <em>(<var>A</var>)</em></dt>
<dd><p>Compute the determinant of <var>A</var>.
</p>
<p>Return an estimate of the reciprocal condition number if requested.
</p>
<p>Programming Notes: Routines from <small>LAPACK</small> are used for full matrices and
code from <small>UMFPACK</small> is used for sparse matrices.
</p>
<p>The determinant should not be used to check a matrix for singularity.
For that, use any of the condition number functions: <code>cond</code>,
<code>condest</code>, <code>rcond</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFcond">cond</a>, <a href="Sparse-Linear-Algebra.html#XREFcondest">condest</a>, <a href="#XREFrcond">rcond</a>.
</p></dd></dl>


<a name="XREFeig"></a><dl>
<dt><a name="index-eig"></a>: <em><var>lambda</var> =</em> <strong>eig</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-eig-1"></a>: <em><var>lambda</var> =</em> <strong>eig</strong> <em>(<var>A</var>, <var>B</var>)</em></dt>
<dt><a name="index-eig-2"></a>: <em>[<var>V</var>, <var>lambda</var>] =</em> <strong>eig</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-eig-3"></a>: <em>[<var>V</var>, <var>lambda</var>] =</em> <strong>eig</strong> <em>(<var>A</var>, <var>B</var>)</em></dt>
<dt><a name="index-eig-4"></a>: <em>[<var>V</var>, <var>lambda</var>, <var>W</var>] =</em> <strong>eig</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-eig-5"></a>: <em>[<var>V</var>, <var>lambda</var>, <var>W</var>] =</em> <strong>eig</strong> <em>(<var>A</var>, <var>B</var>)</em></dt>
<dt><a name="index-eig-6"></a>: <em>[&hellip;] =</em> <strong>eig</strong> <em>(<var>A</var>, <var>balanceOption</var>)</em></dt>
<dt><a name="index-eig-7"></a>: <em>[&hellip;] =</em> <strong>eig</strong> <em>(<var>A</var>, <var>B</var>, <var>algorithm</var>)</em></dt>
<dt><a name="index-eig-8"></a>: <em>[&hellip;] =</em> <strong>eig</strong> <em>(&hellip;, <var>eigvalOption</var>)</em></dt>
<dd><p>Compute the eigenvalues (<var>lambda</var>) and optionally the right eigenvectors
(<var>V</var>) and the left eigenvectors (<var>W</var>) of a matrix or a pair of
matrices.
</p>
<p>The flag <var>balanceOption</var> can be one of:
</p>
<dl compact="compact">
<dt><code>&quot;balance&quot;</code></dt>
<dd><p>Preliminary balancing is on. (default)
</p>
</dd>
<dt><code>&quot;nobalance&quot;</code></dt>
<dd><p>Disables preliminary balancing.
</p></dd>
</dl>

<p>The flag <var>eigvalOption</var> can be one of:
</p>
<dl compact="compact">
<dt><code>&quot;matrix&quot;</code></dt>
<dd><p>Return the eigenvalues in a diagonal matrix. (default if 2 or 3 outputs
are specified)
</p>
</dd>
<dt><code>&quot;vector&quot;</code></dt>
<dd><p>Return the eigenvalues in a column vector. (default if 1 output is
specified, e.g. <var>lambda</var> = eig (<var>A</var>))
</p></dd>
</dl>

<p>The flag <var>algorithm</var> can be one of:
</p>
<dl compact="compact">
<dt><code>&quot;chol&quot;</code></dt>
<dd><p>Uses the Cholesky factorization of B. (default if A is symmetric (Hermitian)
and B is symmetric (Hermitian) positive definite)
</p>
</dd>
<dt><code>&quot;qz&quot;</code></dt>
<dd><p>Uses the QZ algorithm. (When A or B are not symmetric always the
QZ algorithm will be used)
</p></dd>
</dl>

<table>
<thead><tr><th width="31%"></th><th width="23%">no flag</th><th width="23%">chol</th><th width="23%">qz</th></tr></thead>
<tr><td width="31%">both are symmetric</td><td width="23%"><code>&quot;chol&quot;</code></td><td width="23%"><code>&quot;chol&quot;</code></td><td width="23%"><code>&quot;qz&quot;</code></td></tr>
<tr><td width="31%">at least one is not symmetric</td><td width="23%"><code>&quot;qz&quot;</code></td><td width="23%"><code>&quot;qz&quot;</code></td><td width="23%"><code>&quot;qz&quot;</code></td></tr>
</table>

<p>The eigenvalues returned by <code>eig</code> are not ordered.
</p>
<p><strong>See also:</strong> <a href="Sparse-Linear-Algebra.html#XREFeigs">eigs</a>, <a href="Matrix-Factorizations.html#XREFsvd">svd</a>.
</p></dd></dl>


<a name="XREFgivens"></a><dl>
<dt><a name="index-givens"></a>: <em><var>G</var> =</em> <strong>givens</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dt><a name="index-givens-1"></a>: <em>[<var>c</var>, <var>s</var>] =</em> <strong>givens</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dd><p>Compute the Givens rotation matrix <var>G</var>.
</p>
<p>The Givens matrix is a 2 by 2 orthogonal matrix
</p>
<p><code><var>g</var> = [<var>c</var> <var>s</var>; -<var>s</var>' <var>c</var>]</code>
</p>
<p>such that
</p>
<p><code><var>g</var> [<var>x</var>; <var>y</var>] = [*; 0]</code>
</p>
<p>with <var>x</var> and <var>y</var> scalars.
</p>
<p>If two output arguments are requested, return the factors <var>c</var> and
<var>s</var> rather than the Givens rotation matrix.
</p>
<p>For example:
</p>
<div class="example">
<pre class="example">givens (1, 1)
   &rArr;   0.70711   0.70711
       -0.70711   0.70711
</pre></div>

<p><strong>See also:</strong> <a href="#XREFplanerot">planerot</a>.
</p></dd></dl>


<a name="XREFplanerot"></a><dl>
<dt><a name="index-planerot"></a>: <em>[<var>G</var>, <var>y</var>] =</em> <strong>planerot</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Given a two-element column vector, return the
2 by 2 orthogonal matrix
<var>G</var> such that
<code><var>y</var> = <var>g</var> * <var>x</var></code> and <code><var>y</var>(2) = 0</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFgivens">givens</a>.
</p></dd></dl>


<a name="XREFinv"></a><dl>
<dt><a name="index-inv"></a>: <em><var>x</var> =</em> <strong>inv</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-inv-1"></a>: <em>[<var>x</var>, <var>rcond</var>] =</em> <strong>inv</strong> <em>(<var>A</var>)</em></dt>
<dd><p>Compute the inverse of the square matrix <var>A</var>.
</p>
<p>Return an estimate of the reciprocal condition number if requested,
otherwise warn of an ill-conditioned matrix if the reciprocal condition
number is small.
</p>
<p>In general it is best to avoid calculating the inverse of a matrix directly.
For example, it is both faster and more accurate to solve systems of
equations (<var>A</var>*<em>x</em> = <em>b</em>) with
<code><var>y</var> = <var>A</var> \ <em>b</em></code>, rather than
<code><var>y</var> = inv (<var>A</var>) * <em>b</em></code>.
</p>
<p>If called with a sparse matrix, then in general <var>x</var> will be a full
matrix requiring significantly more storage.  Avoid forming the inverse of a
sparse matrix if possible.
</p>
<p><strong>See also:</strong> <a href="Arithmetic-Ops.html#XREFldivide">ldivide</a>, <a href="Arithmetic-Ops.html#XREFrdivide">rdivide</a>, <a href="#XREFpinv">pinv</a>.
</p></dd></dl>


<a name="XREFlinsolve"></a><dl>
<dt><a name="index-linsolve"></a>: <em><var>x</var> =</em> <strong>linsolve</strong> <em>(<var>A</var>, <var>b</var>)</em></dt>
<dt><a name="index-linsolve-1"></a>: <em><var>x</var> =</em> <strong>linsolve</strong> <em>(<var>A</var>, <var>b</var>, <var>opts</var>)</em></dt>
<dt><a name="index-linsolve-2"></a>: <em>[<var>x</var>, <var>R</var>] =</em> <strong>linsolve</strong> <em>(&hellip;)</em></dt>
<dd><p>Solve the linear system <code>A*x = b</code>.
</p>
<p>With no options, this function is equivalent to the left division operator
(<code>x&nbsp;=&nbsp;A&nbsp;\&nbsp;b</code>)<!-- /@w --> or the matrix-left-divide function
(<code>x&nbsp;=&nbsp;mldivide&nbsp;(A,&nbsp;b)</code>)<!-- /@w -->.
</p>
<p>Octave ordinarily examines the properties of the matrix <var>A</var> and chooses
a solver that best matches the matrix.  By passing a structure <var>opts</var>
to <code>linsolve</code> you can inform Octave directly about the matrix <var>A</var>.
In this case Octave will skip the matrix examination and proceed directly
to solving the linear system.
</p>
<p><strong>Warning:</strong> If the matrix <var>A</var> does not have the properties
listed in the <var>opts</var> structure then the result will not be accurate
AND no warning will be given.  When in doubt, let Octave examine the matrix
and choose the appropriate solver as this step takes little time and the
result is cached so that it is only done once per linear system.
</p>
<p>Possible <var>opts</var> fields (set value to true/false):
</p>
<dl compact="compact">
<dt>LT</dt>
<dd><p><var>A</var> is lower triangular
</p>
</dd>
<dt>UT</dt>
<dd><p><var>A</var> is upper triangular
</p>
</dd>
<dt>UHESS</dt>
<dd><p><var>A</var> is upper Hessenberg (currently makes no difference)
</p>
</dd>
<dt>SYM</dt>
<dd><p><var>A</var> is symmetric or complex Hermitian (currently makes no difference)
</p>
</dd>
<dt>POSDEF</dt>
<dd><p><var>A</var> is positive definite
</p>
</dd>
<dt>RECT</dt>
<dd><p><var>A</var> is general rectangular (currently makes no difference)
</p>
</dd>
<dt>TRANSA</dt>
<dd><p>Solve <code>A'*x = b</code> by <code>transpose (A) \ b</code>
</p></dd>
</dl>

<p>The optional second output <var>R</var> is the inverse condition number of
<var>A</var> (zero if matrix is singular).
</p>
<p><strong>See also:</strong> <a href="Arithmetic-Ops.html#XREFmldivide">mldivide</a>, <a href="#XREFmatrix_005ftype">matrix_type</a>, <a href="#XREFrcond">rcond</a>.
</p></dd></dl>


<a name="XREFmatrix_005ftype"></a><dl>
<dt><a name="index-matrix_005ftype"></a>: <em><var>type</var> =</em> <strong>matrix_type</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-matrix_005ftype-1"></a>: <em><var>type</var> =</em> <strong>matrix_type</strong> <em>(<var>A</var>, &quot;nocompute&quot;)</em></dt>
<dt><a name="index-matrix_005ftype-2"></a>: <em><var>A</var> =</em> <strong>matrix_type</strong> <em>(<var>A</var>, <var>type</var>)</em></dt>
<dt><a name="index-matrix_005ftype-3"></a>: <em><var>A</var> =</em> <strong>matrix_type</strong> <em>(<var>A</var>, &quot;upper&quot;, <var>perm</var>)</em></dt>
<dt><a name="index-matrix_005ftype-4"></a>: <em><var>A</var> =</em> <strong>matrix_type</strong> <em>(<var>A</var>, &quot;lower&quot;, <var>perm</var>)</em></dt>
<dt><a name="index-matrix_005ftype-5"></a>: <em><var>A</var> =</em> <strong>matrix_type</strong> <em>(<var>A</var>, &quot;banded&quot;, <var>nl</var>, <var>nu</var>)</em></dt>
<dd><p>Identify the matrix type or mark a matrix as a particular type.
</p>
<p>This allows more rapid solutions of linear equations involving <var>A</var> to be
performed.
</p>
<p>Called with a single argument, <code>matrix_type</code> returns the type of the
matrix and caches it for future use.
</p>
<p>Called with more than one argument, <code>matrix_type</code> allows the type of
the matrix to be defined.
</p>
<p>If the option <code>&quot;nocompute&quot;</code> is given, the function will not attempt
to guess the type if it is still unknown.  This is useful for debugging
purposes.
</p>
<p>The possible matrix types depend on whether the matrix is full or sparse,
and can be one of the following
</p>
<dl compact="compact">
<dt><code>&quot;unknown&quot;</code></dt>
<dd><p>Remove any previously cached matrix type, and mark type as unknown.
</p>
</dd>
<dt><code>&quot;full&quot;</code></dt>
<dd><p>Mark the matrix as full.
</p>
</dd>
<dt><code>&quot;positive definite&quot;</code></dt>
<dd><p>Probable full positive definite matrix.
</p>
</dd>
<dt><code>&quot;diagonal&quot;</code></dt>
<dd><p>Diagonal matrix.  (Sparse matrices only)
</p>
</dd>
<dt><code>&quot;permuted diagonal&quot;</code></dt>
<dd><p>Permuted Diagonal matrix.  The permutation does not need to be specifically
indicated, as the structure of the matrix explicitly gives this.  (Sparse
matrices only)
</p>
</dd>
<dt><code>&quot;upper&quot;</code></dt>
<dd><p>Upper triangular.  If the optional third argument <var>perm</var> is given, the
matrix is assumed to be a permuted upper triangular with the permutations
defined by the vector <var>perm</var>.
</p>
</dd>
<dt><code>&quot;lower&quot;</code></dt>
<dd><p>Lower triangular.  If the optional third argument <var>perm</var> is given, the
matrix is assumed to be a permuted lower triangular with the permutations
defined by the vector <var>perm</var>.
</p>
</dd>
<dt><code>&quot;banded&quot;</code></dt>
<dt><code>&quot;banded positive definite&quot;</code></dt>
<dd><p>Banded matrix with the band size of <var>nl</var> below the diagonal and <var>nu</var>
above it.  If <var>nl</var> and <var>nu</var> are 1, then the matrix is tridiagonal
and treated with specialized code.  In addition the matrix can be marked as
probably a positive definite.  (Sparse matrices only)
</p>
</dd>
<dt><code>&quot;singular&quot;</code></dt>
<dd><p>The matrix is assumed to be singular and will be treated with a minimum norm
solution.
</p>
</dd>
</dl>

<p>Note that the matrix type will be discovered automatically on the first
attempt to solve a linear equation involving <var>A</var>.  Therefore
<code>matrix_type</code> is only useful to give Octave hints of the matrix type.
Incorrectly defining the matrix type will result in incorrect results from
solutions of linear equations; it is entirely <strong>the responsibility of
the user</strong> to correctly identify the matrix type.
</p>
<p>Also, the test for positive definiteness is a low-cost test for a Hermitian
matrix with a real positive diagonal.  This does not guarantee that the
matrix is positive definite, but only that it is a probable candidate.  When
such a matrix is factorized, a Cholesky&nbsp;factorization is first
attempted, and if that fails the matrix is then treated with an
LU&nbsp;factorization.  Once the matrix has been factorized,
<code>matrix_type</code> will return the correct classification of the matrix.
</p></dd></dl>


<a name="XREFnorm"></a><dl>
<dt><a name="index-norm"></a>: <em></em> <strong>norm</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-norm-1"></a>: <em></em> <strong>norm</strong> <em>(<var>A</var>, <var>p</var>)</em></dt>
<dt><a name="index-norm-2"></a>: <em></em> <strong>norm</strong> <em>(<var>A</var>, <var>p</var>, <var>opt</var>)</em></dt>
<dd><p>Compute the p-norm of the matrix <var>A</var>.
</p>
<p>If the second argument is not given, <code>p&nbsp;=&nbsp;2</code><!-- /@w --> is used.
</p>
<p>If <var>A</var> is a matrix (or sparse matrix):
</p>
<dl compact="compact">
<dt><var>p</var> = <code>1</code></dt>
<dd><p>1-norm, the largest column sum of the absolute values of <var>A</var>.
</p>
</dd>
<dt><var>p</var> = <code>2</code></dt>
<dd><p>Largest singular value of <var>A</var>.
</p>
</dd>
<dt><var>p</var> = <code>Inf</code> or <code>&quot;inf&quot;</code></dt>
<dd><a name="index-infinity-norm"></a>
<p>Infinity norm, the largest row sum of the absolute values of <var>A</var>.
</p>
</dd>
<dt><var>p</var> = <code>&quot;fro&quot;</code></dt>
<dd><a name="index-Frobenius-norm"></a>
<p>Frobenius norm of <var>A</var>, <code>sqrt (sum (diag (<var>A</var>' * <var>A</var>)))</code>.
</p>
</dd>
<dt>other <var>p</var>, <code><var>p</var> &gt; 1</code></dt>
<dd><a name="index-general-p_002dnorm"></a>
<p>maximum <code>norm (A*x, p)</code> such that <code>norm (x, p) == 1</code>
</p></dd>
</dl>

<p>If <var>A</var> is a vector or a scalar:
</p>
<dl compact="compact">
<dt><var>p</var> = <code>Inf</code> or <code>&quot;inf&quot;</code></dt>
<dd><p><code>max (abs (<var>A</var>))</code>.
</p>
</dd>
<dt><var>p</var> = <code>-Inf</code></dt>
<dd><p><code>min (abs (<var>A</var>))</code>.
</p>
</dd>
<dt><var>p</var> = <code>&quot;fro&quot;</code></dt>
<dd><p>Frobenius norm of <var>A</var>, <code>sqrt (sumsq (abs (A)))</code>.
</p>
</dd>
<dt><var>p</var> = 0</dt>
<dd><p>Hamming norm&mdash;the number of nonzero elements.
</p>
</dd>
<dt>other <var>p</var>, <code><var>p</var> &gt; 1</code></dt>
<dd><p>p-norm of <var>A</var>, <code>(sum (abs (<var>A</var>) .^ <var>p</var>)) ^ (1/<var>p</var>)</code>.
</p>
</dd>
<dt>other <var>p</var> <code><var>p</var> &lt; 1</code></dt>
<dd><p>the p-pseudonorm defined as above.
</p></dd>
</dl>

<p>If <var>opt</var> is the value <code>&quot;rows&quot;</code>, treat each row as a vector and
compute its norm.  The result is returned as a column vector.
Similarly, if <var>opt</var> is <code>&quot;columns&quot;</code> or <code>&quot;cols&quot;</code> then
compute the norms of each column and return a row vector.
</p>
<p><strong>See also:</strong> <a href="Sparse-Linear-Algebra.html#XREFnormest">normest</a>, <a href="Sparse-Linear-Algebra.html#XREFnormest1">normest1</a>, <a href="#XREFcond">cond</a>, <a href="Matrix-Factorizations.html#XREFsvd">svd</a>.
</p></dd></dl>


<a name="XREFnull"></a><dl>
<dt><a name="index-null"></a>: <em></em> <strong>null</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-null-1"></a>: <em></em> <strong>null</strong> <em>(<var>A</var>, <var>tol</var>)</em></dt>
<dd><p>Return an orthonormal basis of the null space of <var>A</var>.
</p>
<p>The dimension of the null space is taken as the number of singular values of
<var>A</var> not greater than <var>tol</var>.  If the argument <var>tol</var> is missing,
it is computed as
</p>
<div class="example">
<pre class="example">max (size (<var>A</var>)) * max (svd (<var>A</var>)) * eps
</pre></div>

<p><strong>See also:</strong> <a href="#XREForth">orth</a>.
</p></dd></dl>


<a name="XREForth"></a><dl>
<dt><a name="index-orth"></a>: <em></em> <strong>orth</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-orth-1"></a>: <em></em> <strong>orth</strong> <em>(<var>A</var>, <var>tol</var>)</em></dt>
<dd><p>Return an orthonormal basis of the range space of <var>A</var>.
</p>
<p>The dimension of the range space is taken as the number of singular values
of <var>A</var> greater than <var>tol</var>.  If the argument <var>tol</var> is missing, it
is computed as
</p>
<div class="example">
<pre class="example">max (size (<var>A</var>)) * max (svd (<var>A</var>)) * eps
</pre></div>

<p><strong>See also:</strong> <a href="#XREFnull">null</a>.
</p></dd></dl>


<a name="XREFmgorth"></a><dl>
<dt><a name="index-mgorth"></a>: <em>[<var>y</var>, <var>h</var>] =</em> <strong>mgorth</strong> <em>(<var>x</var>, <var>v</var>)</em></dt>
<dd><p>Orthogonalize a given column vector <var>x</var> with respect to a set of
orthonormal vectors comprising the columns of <var>v</var> using the modified
Gram-Schmidt method.
</p>
<p>On exit, <var>y</var> is a unit vector such that:
</p>
<div class="example">
<pre class="example">  norm (<var>y</var>) = 1
  <var>v</var>' * <var>y</var> = 0
  <var>x</var> = [<var>v</var>, <var>y</var>]*<var>h</var>'
</pre></div>

</dd></dl>


<a name="XREFpinv"></a><dl>
<dt><a name="index-pinv"></a>: <em></em> <strong>pinv</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-pinv-1"></a>: <em></em> <strong>pinv</strong> <em>(<var>x</var>, <var>tol</var>)</em></dt>
<dd><p>Return the Moore-Penrose pseudoinverse of <var>x</var>.
</p>
<p>Singular values less than <var>tol</var> are ignored.
</p>
<p>If the second argument is omitted, it is taken to be
</p>
<div class="example">
<pre class="example">tol = max ([rows(<var>x</var>), columns(<var>x</var>)]) * norm (<var>x</var>) * eps
</pre></div>


<p><strong>See also:</strong> <a href="#XREFinv">inv</a>, <a href="Arithmetic-Ops.html#XREFldivide">ldivide</a>.
</p></dd></dl>

<a name="index-pseudoinverse"></a>

<a name="XREFrank"></a><dl>
<dt><a name="index-rank"></a>: <em></em> <strong>rank</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-rank-1"></a>: <em></em> <strong>rank</strong> <em>(<var>A</var>, <var>tol</var>)</em></dt>
<dd><p>Compute the rank of matrix <var>A</var>, using the singular value decomposition.
</p>
<p>The rank is taken to be the number of singular values of <var>A</var> that are
greater than the specified tolerance <var>tol</var>.  If the second argument is
omitted, it is taken to be
</p>
<div class="example">
<pre class="example">tol = max (size (<var>A</var>)) * sigma(1) * eps;
</pre></div>

<p>where <code>eps</code> is machine precision and <code>sigma(1)</code> is the largest
singular value of <var>A</var>.
</p>
<p>The rank of a matrix is the number of linearly independent rows or columns
and determines how many particular solutions exist to a system of equations.
Use <code>null</code> for finding the remaining homogenous solutions.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">x = [1 2 3
     4 5 6
     7 8 9];
rank (x)
  &rArr; 2
</pre></div>

<p>The number of linearly independent rows is only 2 because the final row is a
linear combination of -1*row1 + 2*row2.
</p>

<p><strong>See also:</strong> <a href="#XREFnull">null</a>, <a href="Sparse-Linear-Algebra.html#XREFsprank">sprank</a>, <a href="Matrix-Factorizations.html#XREFsvd">svd</a>.
</p></dd></dl>


<a name="XREFrcond"></a><dl>
<dt><a name="index-rcond"></a>: <em><var>c</var> =</em> <strong>rcond</strong> <em>(<var>A</var>)</em></dt>
<dd><p>Compute the 1-norm estimate of the reciprocal condition number as returned
by <small>LAPACK</small>.
</p>
<p>If the matrix is well-conditioned then <var>c</var> will be near 1 and if the
matrix is poorly conditioned it will be close to 0.
</p>
<p>The matrix <var>A</var> must not be sparse.  If the matrix is sparse then
<code>condest (<var>A</var>)</code> or <code>rcond (full (<var>A</var>))</code> should be used
instead.
</p>
<p><strong>See also:</strong> <a href="#XREFcond">cond</a>, <a href="Sparse-Linear-Algebra.html#XREFcondest">condest</a>.
</p></dd></dl>


<a name="XREFtrace"></a><dl>
<dt><a name="index-trace"></a>: <em></em> <strong>trace</strong> <em>(<var>A</var>)</em></dt>
<dd><p>Compute the trace of <var>A</var>, the sum of the elements along the main
diagonal.
</p>
<p>The implementation is straightforward: <code>sum (diag (<var>A</var>))</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFeig">eig</a>.
</p></dd></dl>


<a name="XREFrref"></a><dl>
<dt><a name="index-rref"></a>: <em></em> <strong>rref</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-rref-1"></a>: <em></em> <strong>rref</strong> <em>(<var>A</var>, <var>tol</var>)</em></dt>
<dt><a name="index-rref-2"></a>: <em>[<var>r</var>, <var>k</var>] =</em> <strong>rref</strong> <em>(&hellip;)</em></dt>
<dd><p>Return the reduced row echelon form of <var>A</var>.
</p>
<p><var>tol</var> defaults to
<code>eps * max (size (<var>A</var>)) * norm (<var>A</var>, inf)</code>.
</p>
<p>The optional return argument <var>k</var> contains the vector of
&quot;bound variables&quot;, which are those columns on which elimination has been
performed.
</p>
</dd></dl>


<hr>
<div class="header">
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