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<h1 class="chapter"> 26. Matrices and Linear Algebra </h1>

<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC86">26.1 Introduction to Matrices and Linear Algebra</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">  
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC90">26.2 Definitions for Matrices and Linear Algebra</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">  
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<h2 class="section"> 26.1 Introduction to Matrices and Linear Algebra </h2>

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<tr><td align="left" valign="top"><a href="#SEC87">26.1.1 Dot</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">                         
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<tr><td align="left" valign="top"><a href="#SEC88">26.1.2 Vectors</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">                     
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<tr><td align="left" valign="top"><a href="#SEC89">26.1.3 eigen</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
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<h3 class="subsection"> 26.1.1 Dot </h3>
<p>The operator <code>.</code> represents noncommutative multiplication and scalar product.
When the operands are 1-column or 1-row matrices <code>a</code> and <code>b</code>,
the expression <code>a.b</code> is equivalent to <code>sum (a[i]*b[i], i, 1, length(a))</code>.
If <code>a</code> and <code>b</code> are not complex, this is the scalar product,
also called the inner product or dot product, of <code>a</code> and <code>b</code>.
The scalar product is defined as <code>conjugate(a).b</code> when <code>a</code> and <code>b</code> are complex;
<code>innerproduct</code> in the <code>eigen</code> package provides the complex scalar product.
</p>
<p>When the operands are more general matrices,
the product is the matrix product <code>a</code> and <code>b</code>.
The number of rows of <code>b</code> must equal the number of columns of <code>a</code>,
and the result has number of rows equal to the number of rows of <code>a</code>
and number of columns equal to the number of columns of <code>b</code>.
</p>
<p>To distinguish <code>.</code> as an arithmetic operator from 
the decimal point in a floating point number,
it may be necessary to leave spaces on either side.
For example, <code>5.e3</code> is <code>5000.0</code> but <code>5 . e3</code> is <code>5</code> times <code>e3</code>.
</p>
<p>There are several flags which govern the simplification of
expressions involving <code>.</code>, namely
<code>dot</code>, <code>dot0nscsimp</code>, <code>dot0simp</code>, <code>dot1simp</code>, <code>dotassoc</code>, 
<code>dotconstrules</code>, <code>dotdistrib</code>, <code>dotexptsimp</code>, <code>dotident</code>,
and <code>dotscrules</code>.
</p>
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<h3 class="subsection"> 26.1.2 Vectors </h3>
<p><code>vect</code> is a package of functions for vector analysis.
<code>load (&quot;vect&quot;)</code> loads this package, and <code>demo (&quot;vect&quot;)</code> displays a demonstration.
</p>
<p>The vector analysis package can combine and simplify symbolic 
expressions including dot products and cross products, together with
the gradient, divergence, curl, and Laplacian operators.  The
distribution of these operators over sums or products is governed
by several flags, as are various other expansions, including expansion
into components in any specific orthogonal coordinate systems.
There are also functions for deriving the scalar or vector potential
of a field.
</p>
<p>The <code>vect</code> package contains these functions:
<code>vectorsimp</code>, <code>scalefactors</code>,
<code>express</code>, <code>potential</code>, and <code>vectorpotential</code>.
</p>
<p>Warning: the <code>vect</code> package declares the dot operator <code>.</code>
to be a commutative operator.
</p>
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<h3 class="subsection"> 26.1.3 eigen </h3>

<p>The package <code>eigen</code> contains several functions devoted to the
symbolic computation of eigenvalues and eigenvectors.
Maxima loads the package automatically if one of the functions
<code>eigenvalues</code> or <code>eigenvectors</code> is invoked.
The package may be loaded explicitly as <code>load (&quot;eigen&quot;)</code>.
</p>
<p><code>demo (&quot;eigen&quot;)</code> displays a demonstration of the capabilities
of this package.
<code>batch (&quot;eigen&quot;)</code> executes the same demonstration,
but without the user prompt between successive computations.
</p>
<p>The functions in the <code>eigen</code> package are
<code>innerproduct</code>, <code>unitvector</code>, <code>columnvector</code>,
<code>gramschmidt</code>, <code>eigenvalues</code>, <code>eigenvectors</code>, <code>uniteigenvectors</code>,
and <code>similaritytransform</code>.
</p>
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<h2 class="section"> 26.2 Definitions for Matrices and Linear Algebra </h2>

<dl>
<dt><u>Function:</u> <b>addcol</b><i> (<var>M</var>, <var>list_1</var>, ..., <var>list_n</var>)</i>
<a name="IDX751"></a>
</dt>
<dd><p>Appends the column(s) given by the one
or more lists (or matrices) onto the matrix <var>M</var>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>addrow</b><i> (<var>M</var>, <var>list_1</var>, ..., <var>list_n</var>)</i>
<a name="IDX752"></a>
</dt>
<dd><p>Appends the row(s) given by the one or
more lists (or matrices) onto the matrix <var>M</var>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>adjoint</b><i> (<var>M</var>)</i>
<a name="IDX753"></a>
</dt>
<dd><p>Returns the adjoint of the matrix <var>M</var>.
The adjoint matrix is the transpose of the matrix of cofactors of <var>M</var>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>augcoefmatrix</b><i> ([<var>eqn_1</var>, ..., <var>eqn_m</var>], [<var>x_1</var>, ..., <var>x_n</var>])</i>
<a name="IDX754"></a>
</dt>
<dd><p>Returns the augmented coefficient
matrix for the variables <var>x_1</var>, ..., <var>x_n</var> of the system of linear equations
<var>eqn_1</var>, ..., <var>eqn_m</var>.  This is the coefficient matrix with a column adjoined for
the constant terms in each equation (i.e., those terms not dependent upon
<var>x_1</var>, ..., <var>x_n</var>).
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) m: [2*x - (a - 1)*y = 5*b, c + b*y + a*x = 0]$
(%i2) augcoefmatrix (m, [x, y]);
                       [ 2  1 - a  - 5 b ]
(%o2)                  [                 ]
                       [ a    b      c   ]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>charpoly</b><i> (<var>M</var>, <var>x</var>)</i>
<a name="IDX755"></a>
</dt>
<dd><p>Returns the characteristic polynomial for the matrix <var>M</var>
with respect to variable <var>x</var>.  That is,
<code>determinant (<var>M</var> - diagmatrix (length (<var>M</var>), <var>x</var>))</code>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) a: matrix ([3, 1], [2, 4]);
                            [ 3  1 ]
(%o1)                       [      ]
                            [ 2  4 ]
(%i2) expand (charpoly (a, lambda));
                           2
(%o2)                lambda  - 7 lambda + 10
(%i3) (programmode: true, solve (%));
(%o3)               [lambda = 5, lambda = 2]
(%i4) matrix ([x1], [x2]);
                             [ x1 ]
(%o4)                        [    ]
                             [ x2 ]
(%i5) ev (a . % - lambda*%, %th(2)[1]);
                          [ x2 - 2 x1 ]
(%o5)                     [           ]
                          [ 2 x1 - x2 ]
(%i6) %[1, 1] = 0;
(%o6)                     x2 - 2 x1 = 0
(%i7) x2^2 + x1^2 = 1;
                            2     2
(%o7)                     x2  + x1  = 1
(%i8) solve ([%th(2), %], [x1, x2]);
                  1               2
(%o8) [[x1 = - -------, x2 = - -------], 
               sqrt(5)         sqrt(5)

                                             1             2
                                    [x1 = -------, x2 = -------]]
                                          sqrt(5)       sqrt(5)
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>coefmatrix</b><i> ([<var>eqn_1</var>, ..., <var>eqn_m</var>], [<var>x_1</var>, ..., <var>x_n</var>])</i>
<a name="IDX756"></a>
</dt>
<dd><p>Returns the coefficient matrix for the
variables <var>x_1</var>, ..., <var>x_n</var> of the system of linear equations 
<var>eqn_1</var>, ..., <var>eqn_m</var>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) coefmatrix([2*x-(a-1)*y+5*b = 0, b*y+a*x = 3], [x,y]);
                                 [ 2  1 - a ]
(%o1)                            [          ]
                                 [ a    b   ]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>col</b><i> (<var>M</var>, <var>i</var>)</i>
<a name="IDX757"></a>
</dt>
<dd><p>Returns the <var>i</var>'th column of the matrix <var>M</var>.
The return value is a matrix.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>columnvector</b><i> (<var>L</var>)</i>
<a name="IDX758"></a>
</dt>
<dt><u>Function:</u> <b>covect</b><i> (<var>L</var>)</i>
<a name="IDX759"></a>
</dt>
<dd><p>Returns a matrix of one column and <code>length (<var>L</var>)</code> rows,
containing the elements of the list <var>L</var>.
</p>
<p><code>covect</code> is a synonym for <code>columnvector</code>.
</p>
<p><code>load (&quot;eigen&quot;)</code> loads this function.
</p>
<p>This is useful if you want to use parts of the outputs of
the functions in this package in matrix calculations.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (&quot;eigen&quot;)$
Warning - you are redefining the Macsyma function eigenvalues
Warning - you are redefining the Macsyma function eigenvectors
(%i2) columnvector ([aa, bb, cc, dd]);
                             [ aa ]
                             [    ]
                             [ bb ]
(%o2)                        [    ]
                             [ cc ]
                             [    ]
                             [ dd ]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>conjugate</b><i> (<var>x</var>)</i>
<a name="IDX760"></a>
</dt>
<dd><p>Returns the complex conjugate of <var>x</var>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) declare ([aa, bb], real, cc, complex, ii, imaginary);

(%o1)                         done
(%i2) conjugate (aa + bb*%i);

(%o2)                      aa - %i bb
(%i3) conjugate (cc);

(%o3)                     conjugate(cc)
(%i4) conjugate (ii);

(%o4)                         - ii
(%i5) conjugate (xx + yy);

(%o5)             conjugate(yy) + conjugate(xx)
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>copymatrix</b><i> (<var>M</var>)</i>
<a name="IDX761"></a>
</dt>
<dd><p>Returns a copy of the matrix <var>M</var>.  This is the only way
to make a copy aside from copying <var>M</var> element by element.
</p>
<p>Note that an assignment of one matrix to another, as in <code>m2: m1</code>,
does not copy <code>m1</code>.
An assignment <code>m2 [i,j]: x</code> or <code>setelmx (x, i, j, m2</code> also modifies <code>m1 [i,j]</code>.
Creating a copy with <code>copymatrix</code> and then using assignment creates a separate, modified copy.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>determinant</b><i> (<var>M</var>)</i>
<a name="IDX762"></a>
</dt>
<dd><p>Computes the determinant of <var>M</var> by a method similar to
Gaussian elimination.
</p>
<p>The form of the result depends upon the setting
of the switch <code>ratmx</code>.
</p>
<p>There is a special routine for computing
sparse determinants which is called when the switches
<code>ratmx</code> and <code>sparse</code> are both <code>true</code>.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>detout</b>
<a name="IDX763"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>detout</code> is <code>true</code>, the determinant of a
matrix whose inverse is computed is factored out of the inverse.
</p>
<p>For this switch to have an effect <code>doallmxops</code> and <code>doscmxops</code> should be
<code>false</code> (see their descriptions).  Alternatively this switch can be
given to <code>ev</code> which causes the other two to be set correctly.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) m: matrix ([a, b], [c, d]);
                            [ a  b ]
(%o1)                       [      ]
                            [ c  d ]
(%i2) detout: true$
(%i3) doallmxops: false$
(%i4) doscmxops: false$
(%i5) invert (m);
                          [  d   - b ]
                          [          ]
                          [ - c   a  ]
(%o5)                     ------------
                           a d - b c
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>diagmatrix</b><i> (<var>n</var>, <var>x</var>)</i>
<a name="IDX764"></a>
</dt>
<dd><p>Returns a diagonal matrix of size <var>n</var> by <var>n</var> with the
diagonal elements all equal to <var>x</var>.
<code>diagmatrix (<var>n</var>, 1)</code> returns an identity matrix (same as <code>ident (<var>n</var>)</code>).
</p>
<p><var>n</var> must evaluate to an integer, otherwise <code>diagmatrix</code> complains with an error message.
</p>
<p><var>x</var> can be any kind of expression, including another matrix.
If <var>x</var> is a matrix, it is not copied; all diagonal elements refer to the same instance, <var>x</var>.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>doallmxops</b>
<a name="IDX765"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>doallmxops</code> is <code>true</code>,
all operations relating to matrices are carried out.
When it is <code>false</code> then the setting of the
individual <code>dot</code> switches govern which operations are performed.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>domxexpt</b>
<a name="IDX766"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>domxexpt</code> is <code>true</code>,
a matrix exponential, <code>exp (<var>M</var>)</code> where <var>M</var> is a matrix,
is interpreted as a matrix with element <code>[i,j</code> equal to <code>exp (m[i,j])</code>.
Otherwise <code>exp (<var>M</var>)</code> evaluates to <code>exp (<var>ev(M)</var></code>.
</p>
<p><code>domxexpt</code>
affects all expressions of the form <code><var>base</var>^<var>power</var></code> where <var>base</var> is an
expression assumed scalar or constant, and <var>power</var> is a list or
matrix.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) m: matrix ([1, %i], [a+b, %pi]);
                         [   1    %i  ]
(%o1)                    [            ]
                         [ b + a  %pi ]
(%i2) domxexpt: false$
(%i3) (1 - c)^m;
                             [   1    %i  ]
                             [            ]
                             [ b + a  %pi ]
(%o3)                 (1 - c)
(%i4) domxexpt: true$
(%i5) (1 - c)^m;
                  [                      %i  ]
                  [    1 - c      (1 - c)    ]
(%o5)             [                          ]
                  [        b + a         %pi ]
                  [ (1 - c)       (1 - c)    ]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>domxmxops</b>
<a name="IDX767"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>domxmxops</code> is <code>true</code>, all matrix-matrix or
matrix-list operations are carried out (but not scalar-matrix
operations); if this switch is <code>false</code> such operations are not carried out.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>domxnctimes</b>
<a name="IDX768"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>domxnctimes</code> is <code>true</code>, non-commutative products of
matrices are carried out.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>dontfactor</b>
<a name="IDX769"></a>
</dt>
<dd><p>Default value: <code>[]</code>
</p>
<p><code>dontfactor</code> may be set to a list of variables with respect
to which factoring is not to occur.  (The list is initially empty.)
Factoring also will not take place with respect to any variables which
are less important, according the variable ordering assumed for canonical rational expression (CRE) form,
than those on the <code>dontfactor</code> list.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>doscmxops</b>
<a name="IDX770"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>doscmxops</code> is <code>true</code>, scalar-matrix operations are
carried out.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>doscmxplus</b>
<a name="IDX771"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>doscmxplus</code> is <code>true</code>, scalar-matrix operations yield
a matrix result.  This switch is not subsumed under <code>doallmxops</code>.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>dot0nscsimp</b>
<a name="IDX772"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>dot0nscsimp</code> is <code>true</code>, a non-commutative product of zero
and a nonscalar term is simplified to a commutative product.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>dot0simp</b>
<a name="IDX773"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>dot0simp</code> is <code>true</code>,
a non-commutative product of zero and
a scalar term is simplified to a commutative product.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>dot1simp</b>
<a name="IDX774"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>dot1simp</code> is <code>true</code>,
a non-commutative product of one and
another term is simplified to a commutative product.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>dotassoc</b>
<a name="IDX775"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>dotassoc</code> is <code>true</code>, an expression <code>(A.B).C</code> simplifies to
<code>A.(B.C)</code>.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>dotconstrules</b>
<a name="IDX776"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>dotconstrules</code> is <code>true</code>, a non-commutative product of a
constant and another term is simplified to a commutative product.
Turning on this flag effectively turns on <code>dot0simp</code>, <code>dot0nscsimp</code>, and
<code>dot1simp</code> as well.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>dotdistrib</b>
<a name="IDX777"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>dotdistrib</code> is <code>true</code>, an expression <code>A.(B + C)</code> simplifies to <code>A.B + A.C</code>.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>dotexptsimp</b>
<a name="IDX778"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>dotexptsimp</code> is <code>true</code>, an expression <code>A.A</code> simplifies to <code>A^^2</code>.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>dotident</b>
<a name="IDX779"></a>
</dt>
<dd><p>Default value: 1
</p>
<p><code>dotident</code> is the value returned by <code>X^^0</code>.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>dotscrules</b>
<a name="IDX780"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>dotscrules</code> is <code>true</code>, an expression <code>A.SC</code> or <code>SC.A</code> simplifies
to <code>SC*A</code> and <code>A.(SC*B)</code> simplifies to <code>SC*(A.B)</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>echelon</b><i> (<var>M</var>)</i>
<a name="IDX781"></a>
</dt>
<dd><p>Returns the echelon form of the matrix <var>M</var>,
as produced by Gaussian elimination.
The echelon form is computed from <var>M</var>
by elementary row operations such that the first
non-zero element in each row in the resulting matrix is one and the
column elements under the first one in each row are all zero.
</p>
<p><code>triangularize</code> also carries out Gaussian elimination,
but it does not normalize the leading non-zero element in each row.
</p>
<p><code>lu_factor</code> and <code>cholesky</code> are other functions which yield triangularized matrices.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) M: matrix ([3, 7, aa, bb], [-1, 8, 5, 2], [9, 2, 11, 4]);
                       [  3   7  aa  bb ]
                       [                ]
(%o1)                  [ - 1  8  5   2  ]
                       [                ]
                       [  9   2  11  4  ]
(%i2) echelon (M);
                  [ 1  - 8  - 5      - 2     ]
                  [                          ]
                  [         28       11      ]
                  [ 0   1   --       --      ]
(%o2)             [         37       37      ]
                  [                          ]
                  [              37 bb - 119 ]
                  [ 0   0    1   ----------- ]
                  [              37 aa - 313 ]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>eigenvalues</b><i> (<var>M</var>)</i>
<a name="IDX782"></a>
</dt>
<dt><u>Function:</u> <b>eivals</b><i> (<var>M</var>)</i>
<a name="IDX783"></a>
</dt>
<dd><p>Returns a list of two lists containing the eigenvalues of the matrix <var>M</var>.
The first sublist of the return value is the list of eigenvalues of the
matrix, and the second sublist is the list of the
multiplicities of the eigenvalues in the corresponding order.
</p>
<p><code>eivals</code> is a synonym for <code>eigenvalues</code>.
</p>
<p><code>eigenvalues</code> calls the function <code>solve</code> to find the roots of the
characteristic polynomial of the matrix.
Sometimes <code>solve</code> may not be able to find the roots of the polynomial;
in that case some other functions in this
package (except <code>innerproduct</code>, <code>unitvector</code>, <code>columnvector</code> and
<code>gramschmidt</code>) will not work.
</p>
<p>In some cases the eigenvalues found by <code>solve</code> may be complicated expressions.
(This may happen when <code>solve</code> returns a not-so-obviously real expression
for an eigenvalue which is known to be real.)
It may be possible to simplify the eigenvalues using some other functions.
</p>
<p>The package <code>eigen.mac</code> is loaded automatically when
<code>eigenvalues</code> or <code>eigenvectors</code> is referenced.
If <code>eigen.mac</code> is not already loaded,
<code>load (&quot;eigen&quot;)</code> loads it.
After loading, all functions and variables in the package are available.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>eigenvectors</b><i> (<var>M</var>)</i>
<a name="IDX784"></a>
</dt>
<dt><u>Function:</u> <b>eivects</b><i> (<var>M</var>)</i>
<a name="IDX785"></a>
</dt>
<dd><p>takes a matrix <var>M</var> as its argument and returns a list
of lists the first sublist of which is the output of <code>eigenvalues</code>
and the other sublists of which are the eigenvectors of the
matrix corresponding to those eigenvalues respectively.
The calculated eigenvectors and the unit eigenvectors of the matrix are the
right eigenvectors and the right unit eigenvectors respectively.
</p>
<p><code>eivects</code> is a synonym for <code>eigenvectors</code>.
</p>
<p>The package <code>eigen.mac</code> is loaded automatically when
<code>eigenvalues</code> or <code>eigenvectors</code> is referenced.
If <code>eigen.mac</code> is not already loaded,
<code>load (&quot;eigen&quot;)</code> loads it.
After loading, all functions and variables in the package are available.
</p>
<p>The flags that affect this function are:
</p>
<p><code>nondiagonalizable</code> is set to <code>true</code> or <code>false</code> depending on
whether the matrix is nondiagonalizable or diagonalizable after
<code>eigenvectors</code> returns.
</p>
<p><code>hermitianmatrix</code> when <code>true</code>, causes the degenerate
eigenvectors of the Hermitian matrix to be orthogonalized using the
Gram-Schmidt algorithm.
</p>
<p><code>knowneigvals</code> when <code>true</code> causes the <code>eigen</code> package to assume the
eigenvalues of the matrix are known to the user and stored under the
global name <code>listeigvals</code>.  <code>listeigvals</code> should be set to a list similar
to the output <code>eigenvalues</code>.
</p>
<p>The function <code>algsys</code> is used here to solve for the eigenvectors. Sometimes if the
eigenvalues are messy, <code>algsys</code> may not be able to find a solution.
In some cases, it may be possible to simplify the eigenvalues by
first finding them using <code>eigenvalues</code> command and then using other functions
to reduce them to something simpler.
Following simplification, <code>eigenvectors</code> can be called again
with the <code>knowneigvals</code> flag set to <code>true</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>ematrix</b><i> (<var>m</var>, <var>n</var>, <var>x</var>, <var>i</var>, <var>j</var>)</i>
<a name="IDX786"></a>
</dt>
<dd><p>Returns an <var>m</var> by <var>n</var> matrix, all elements of which
are zero except for the <code>[<var>i</var>, <var>j</var>]</code> element which is <var>x</var>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>entermatrix</b><i> (<var>m</var>, <var>n</var>)</i>
<a name="IDX787"></a>
</dt>
<dd><p>Returns an <var>m</var> by <var>n</var> matrix, reading the elements interactively.
</p>
<p>If <var>n</var> is equal to <var>m</var>,
Maxima prompts for the type of the matrix (diagonal, symmetric, antisymmetric, or general)
and for each element.
Each response is terminated by a semicolon <code>;</code> or dollar sign <code>$</code>.
</p>
<p>If <var>n</var> is not equal to <var>m</var>,
Maxima prompts for each element.
</p>
<p>The elements may be any expressions, which are evaluated.
<code>entermatrix</code> evaluates its arguments.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) n: 3$
(%i2) m: entermatrix (n, n)$

Is the matrix  1. Diagonal  2. Symmetric  3. Antisymmetric  4. General
Answer 1, 2, 3 or 4 : 
1$
Row 1 Column 1: 
(a+b)^n$
Row 2 Column 2: 
(a+b)^(n+1)$
Row 3 Column 3: 
(a+b)^(n+2)$

Matrix entered.
(%i3) m;
                [        3                     ]
                [ (b + a)      0         0     ]
                [                              ]
(%o3)           [                  4           ]
                [    0      (b + a)      0     ]
                [                              ]
                [                            5 ]
                [    0         0      (b + a)  ]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>genmatrix</b><i> (<var>a</var>, <var>i_2</var>, <var>j_2</var>, <var>i_1</var>, <var>j_1</var>)</i>
<a name="IDX788"></a>
</dt>
<dt><u>Function:</u> <b>genmatrix</b><i> (<var>a</var>, <var>i_2</var>, <var>j_2</var>, <var>i_1</var>)</i>
<a name="IDX789"></a>
</dt>
<dt><u>Function:</u> <b>genmatrix</b><i> (<var>a</var>, <var>i_2</var>, <var>j_2</var>)</i>
<a name="IDX790"></a>
</dt>
<dd><p>Returns a matrix generated from <var>a</var>,
taking element <code><var>a</var>[<var>i_1</var>,<var>j_1</var>]</code>
as the upper-left element and <code><var>a</var>[<var>i_2</var>,<var>j_2</var>]</code>
as the lower-right element of the matrix.
Here <var>a</var> is a declared array (created by <code>array</code> but not by <code>make_array</code>)
or an undeclared array,
or an array function,
or a lambda expression of two arguments.
(An array function is created like other functions with <code>:=</code> or <code>define</code>,
but arguments are enclosed in square brackets instead of parentheses.)
</p>
<p>If <var>j_1</var> is omitted, it is assumed equal to <var>i_1</var>.
If both <var>j_1</var> and <var>i_1</var> are omitted, both are assumed equal to 1.
</p>
<p>If a selected element <code>i,j</code> of the array is undefined,
the matrix will contain a symbolic element <code><var>a</var>[i,j]</code>.
</p>
<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) h [i, j] := 1 / (i + j - 1);
                                    1
(%o1)                  h     := ---------
                        i, j    i + j - 1
(%i2) genmatrix (h, 3, 3);
                           [    1  1 ]
                           [ 1  -  - ]
                           [    2  3 ]
                           [         ]
                           [ 1  1  1 ]
(%o2)                      [ -  -  - ]
                           [ 2  3  4 ]
                           [         ]
                           [ 1  1  1 ]
                           [ -  -  - ]
                           [ 3  4  5 ]
(%i3) array (a, fixnum, 2, 2);
(%o3)                           a
(%i4) a [1, 1] : %e;
(%o4)                          %e
(%i5) a [2, 2] : %pi;
(%o5)                          %pi
(%i6) genmatrix (a, 2, 2);
                           [ %e   0  ]
(%o6)                      [         ]
                           [ 0   %pi ]
(%i7) genmatrix (lambda ([i, j], j - i), 3, 3);
                         [  0    1   2 ]
                         [             ]
(%o7)                    [ - 1   0   1 ]
                         [             ]
                         [ - 2  - 1  0 ]
(%i8) genmatrix (B, 2, 2);
                        [ B      B     ]
                        [  1, 1   1, 2 ]
(%o8)                   [              ]
                        [ B      B     ]
                        [  2, 1   2, 2 ]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>gramschmidt</b><i> (<var>x</var>)</i>
<a name="IDX791"></a>
</dt>
<dt><u>Function:</u> <b>gschmit</b><i> (<var>x</var>)</i>
<a name="IDX792"></a>
</dt>
<dd><p>Carries out the Gram-Schmidt orthogonalization algorithm on <var>x</var>,
which is either a matrix or a list of lists.
<var>x</var> is not modified by <code>gramschmidt</code>.
</p>
<p>If <var>x</var> is a matrix, the algorithm is applied to the rows of <var>x</var>.
If <var>x</var> is a list of lists, the algorithm is applied to the sublists,
which must have equal numbers of elements.
In either case,
the return value is a list of lists, the sublists of which are orthogonal
and span the same space as <var>x</var>.
If the dimension of the span of <var>x</var> is less than the number of rows or sublists,
some sublists of the return value are zero.
</p>
<p><code>factor</code> is called at each stage of the algorithm to simplify intermediate results.
As a consequence, the return value may contain factored integers.
</p>
<p><code>gschmit</code> (note spelling) is a synonym for <code>gramschmidt</code>.
</p>
<p><code>load (&quot;eigen&quot;)</code> loads this function.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (&quot;eigen&quot;)$
Warning - you are redefining the Macsyma function eigenvalues
Warning - you are redefining the Macsyma function eigenvectors
(%i2) x: matrix ([1, 2, 3], [9, 18, 30], [12, 48, 60]);
                         [ 1   2   3  ]
                         [            ]
(%o2)                    [ 9   18  30 ]
                         [            ]
                         [ 12  48  60 ]
(%i3) y: gramschmidt (x);
                       2      2            4     3
                      3      3   3 5      2  3  2  3
(%o3)  [[1, 2, 3], [- ---, - --, ---], [- ----, ----, 0]]
                      2 7    7   2 7       5     5
(%i4) i: innerproduct$
(%i5) [i (y[1], y[2]), i (y[2], y[3]), i (y[3], y[1])];
(%o5)                       [0, 0, 0]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>hach</b><i> (<var>a</var>, <var>b</var>, <var>m</var>, <var>n</var>, <var>l</var>)</i>
<a name="IDX793"></a>
</dt>
<dd><p><code>hach</code> is an implementation of Hacijan's linear programming algorithm.
</p>
<p><code>load (&quot;kach&quot;)</code> loads this function.
<code>demo (&quot;kach&quot;)</code> executes a demonstration of this function.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>ident</b><i> (<var>n</var>)</i>
<a name="IDX794"></a>
</dt>
<dd><p>Returns an <var>n</var> by <var>n</var> identity matrix.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>innerproduct</b><i> (<var>x</var>, <var>y</var>)</i>
<a name="IDX795"></a>
</dt>
<dt><u>Function:</u> <b>inprod</b><i> (<var>x</var>, <var>y</var>)</i>
<a name="IDX796"></a>
</dt>
<dd><p>Returns the inner product (also called the scalar product or dot product) of <var>x</var> and <var>y</var>,
which are lists of equal length, or both 1-column or 1-row matrices of equal length.
The return value is <code>conjugate (x) . y</code>,
where <code>.</code> is the noncommutative multiplication operator.
</p>
<p><code>load (&quot;eigen&quot;)</code> loads this function.
</p>
<p><code>inprod</code> is a synonym for <code>innerproduct</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>invert</b><i> (<var>M</var>)</i>
<a name="IDX797"></a>
</dt>
<dd><p>Returns the inverse of the matrix <var>M</var>.
The inverse is computed by the adjoint method.
</p>
<p>This allows a user to compute the inverse of a matrix with
bfloat entries or polynomials with floating pt. coefficients without
converting to cre-form.
</p>
<p>Cofactors are computed by the <code>determinant</code> function,
so if <code>ratmx</code> is <code>false</code> the inverse is computed
without changing the representation of the elements.
</p>
<p>The current
implementation is inefficient for matrices of high order.
</p>
<p>When <code>detout</code> is <code>true</code>, the determinant is factored out of the
inverse.
</p>
<p>The elements of the inverse are not automatically expanded.
If <var>M</var> has polynomial elements, better appearing output can be
generated by <code>expand (invert (m)), detout</code>.
If it is desirable to then
divide through by the determinant this can be accomplished by <code>xthru (%)</code>
or alternatively from scratch by
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">expand (adjoint (m)) / expand (determinant (m))
invert (m) := adjoint (m) / determinant (m)
</pre></td></tr></table>
<p>See <code>^^</code> (noncommutative exponent) for another method of inverting a matrix.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>lmxchar</b>
<a name="IDX798"></a>
</dt>
<dd><p>Default value: <code>[</code>
</p>
<p><code>lmxchar</code> is the character displayed as the left
delimiter of a matrix.
See also <code>rmxchar</code>.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) lmxchar: &quot;|&quot;$
(%i2) matrix ([a, b, c], [d, e, f], [g, h, i]);
                           | a  b  c ]
                           |         ]
(%o2)                      | d  e  f ]
                           |         ]
                           | g  h  i ]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>matrix</b><i> (<var>row_1</var>, ..., <var>row_n</var>)</i>
<a name="IDX799"></a>
</dt>
<dd><p>Returns a rectangular matrix which has the rows <var>row_1</var>, ..., <var>row_n</var>.
Each row is a list of expressions.
All rows must be the same length.
</p>
<p>The operations <code>+</code> (addition), <code>-</code> (subtraction), <code>*</code> (multiplication),
and <code>/</code> (division), are carried out element by element
when the operands are two matrices, a scalar and a matrix, or a matrix and a scalar.
The operation <code>^</code> (exponentiation, equivalently <code>**</code>)
is carried out element by element 
if the operands are a scalar and a matrix or a matrix and a scalar,
but not if the operands are two matrices.
All operations are normally carried out in full,
including <code>.</code> (noncommutative multiplication).
</p>
<p>Matrix multiplication is represented by the noncommutative multiplication operator <code>.</code>.
The corresponding noncommutative exponentiation operator is <code>^^</code>.
For a matrix <code><var>A</var></code>, <code><var>A</var>.<var>A</var> = <var>A</var>^^2</code> and
<code><var>A</var>^^-1</code> is the inverse of <var>A</var>, if it exists.
</p>
<p>There are switches for controlling simplification of expressions
involving dot and matrix-list operations.
These are
<code>doallmxops</code>, <code>domxexpt</code>
<code>domxmxops</code>, <code>doscmxops</code>, and <code>doscmxplus</code>.
</p>
<p>There are additional options which are related to matrices. These are:
<code>lmxchar</code>, <code>rmxchar</code>, <code>ratmx</code>, <code>listarith</code>, <code>detout</code>,
<code>scalarmatrix</code>,
and <code>sparse</code>.
</p>
<p>There are a number of
functions which take matrices as arguments or yield matrices as return values.
See <code>eigenvalues</code>, <code>eigenvectors</code>,
<code>determinant</code>,
<code>charpoly</code>, <code>genmatrix</code>, <code>addcol</code>, <code>addrow</code>, 
<code>copymatrix</code>, <code>transpose</code>, <code>echelon</code>,
and <code>rank</code>.
</p>
<p>Examples:
</p>
<ul>
<li>
Construction of matrices from lists.
</li></ul>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) x: matrix ([17, 3], [-8, 11]);
                           [ 17   3  ]
(%o1)                      [         ]
                           [ - 8  11 ]
(%i2) y: matrix ([%pi, %e], [a, b]);
                           [ %pi  %e ]
(%o2)                      [         ]
                           [  a   b  ]
</pre></td></tr></table><ul>
<li>
Addition, element by element.
</li></ul>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i3) x + y;
                      [ %pi + 17  %e + 3 ]
(%o3)                 [                  ]
                      [  a - 8    b + 11 ]
</pre></td></tr></table><ul>
<li>
Subtraction, element by element.
</li></ul>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i4) x - y;
                      [ 17 - %pi  3 - %e ]
(%o4)                 [                  ]
                      [ - a - 8   11 - b ]
</pre></td></tr></table><ul>
<li>
Multiplication, element by element.
</li></ul>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i5) x * y;
                        [ 17 %pi  3 %e ]
(%o5)                   [              ]
                        [ - 8 a   11 b ]
</pre></td></tr></table><ul>
<li>
Division, element by element.
</li></ul>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i6) x / y;
                        [ 17       - 1 ]
                        [ ---  3 %e    ]
                        [ %pi          ]
(%o6)                   [              ]
                        [   8    11    ]
                        [ - -    --    ]
                        [   a    b     ]
</pre></td></tr></table><ul>
<li>
Matrix to a scalar exponent, element by element.
</li></ul>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i7) x ^ 3;
                         [ 4913    27  ]
(%o7)                    [             ]
                         [ - 512  1331 ]
</pre></td></tr></table><ul>
<li>
Scalar base to a matrix exponent, element by element.
</li></ul>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i8) exp(y); 
                         [   %pi    %e ]
                         [ %e     %e   ]
(%o8)                    [             ]
                         [    a     b  ]
                         [  %e    %e   ]
</pre></td></tr></table><ul>
<li>
Matrix base to a matrix exponent. This is not carried out element by element.
</li></ul>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i9) x ^ y;
                                [ %pi  %e ]
                                [         ]
                                [  a   b  ]
                     [ 17   3  ]
(%o9)                [         ]
                     [ - 8  11 ]
</pre></td></tr></table><ul>
<li>
Noncommutative matrix multiplication.
</li></ul>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i10) x . y;
                  [ 3 a + 17 %pi  3 b + 17 %e ]
(%o10)            [                           ]
                  [ 11 a - 8 %pi  11 b - 8 %e ]
(%i11) y . x;
                [ 17 %pi - 8 %e  3 %pi + 11 %e ]
(%o11)          [                              ]
                [  17 a - 8 b     11 b + 3 a   ]
</pre></td></tr></table><ul>
<li>
Noncommutative matrix exponentiation.
A scalar base <var>b</var> to a matrix power <var>M</var>
is carried out element by element and so <code>b^^m</code> is the same as <code>b^m</code>.
</li></ul>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i12) x ^^ 3;
                        [  3833   1719 ]
(%o12)                  [              ]
                        [ - 4584  395  ]
(%i13) %e ^^ y;
                         [   %pi    %e ]
                         [ %e     %e   ]
(%o13)                   [             ]
                         [    a     b  ]
                         [  %e    %e   ]
</pre></td></tr></table><ul>
<li>
A matrix raised to a -1 exponent with noncommutative exponentiation is the matrix inverse,
if it exists.
</li></ul>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i14) x ^^ -1;
                         [ 11      3  ]
                         [ ---  - --- ]
                         [ 211    211 ]
(%o14)                   [            ]
                         [  8    17   ]
                         [ ---   ---  ]
                         [ 211   211  ]
(%i15) x . (x ^^ -1);
                            [ 1  0 ]
(%o15)                      [      ]
                            [ 0  1 ]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>matrixmap</b><i> (<var>f</var>, <var>M</var>)</i>
<a name="IDX800"></a>
</dt>
<dd><p>Returns a matrix with element <code>i,j</code> equal to <code><var>f</var>(<var>M</var>[i,j])</code>.
</p>
<p>See also <code>map</code>, <code>fullmap</code>, <code>fullmapl</code>, and <code>apply</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>matrixp</b><i> (<var>expr</var>)</i>
<a name="IDX801"></a>
</dt>
<dd><p>Returns <code>true</code> if <var>expr</var> is a matrix, otherwise <code>false</code>.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>matrix_element_add</b>
<a name="IDX802"></a>
</dt>
<dd><p>Default value: <code>+</code>
</p>
<p><code>matrix_element_add</code> is the operation 
invoked in place of addition in a matrix multiplication.
<code>matrix_element_add</code> can be assigned any n-ary operator
(that is, a function which handles any number of arguments).
The assigned value may be the name of an operator enclosed in quote marks,
the name of a function,
or a lambda expression.
</p>
<p>See also <code>matrix_element_mult</code> and <code>matrix_element_transpose</code>.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) matrix_element_add: &quot;*&quot;$
(%i2) matrix_element_mult: &quot;^&quot;$
(%i3) aa: matrix ([a, b, c], [d, e, f]);
                           [ a  b  c ]
(%o3)                      [         ]
                           [ d  e  f ]
(%i4) bb: matrix ([u, v, w], [x, y, z]);
                           [ u  v  w ]
(%o4)                      [         ]
                           [ x  y  z ]
(%i5) aa . transpose (bb);
                     [  u  v  w   x  y  z ]
                     [ a  b  c   a  b  c  ]
(%o5)                [                    ]
                     [  u  v  w   x  y  z ]
                     [ d  e  f   d  e  f  ]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>matrix_element_mult</b>
<a name="IDX803"></a>
</dt>
<dd><p>Default value: <code>*</code>
</p>
<p><code>matrix_element_mult</code> is the operation 
invoked in place of multiplication in a matrix multiplication.
<code>matrix_element_mult</code> can be assigned any binary operator.
The assigned value may be the name of an operator enclosed in quote marks,
the name of a function,
or a lambda expression.
</p>
<p>The dot operator <code>.</code> is a useful choice in some contexts.
</p>
<p>See also <code>matrix_element_add</code> and <code>matrix_element_transpose</code>.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) matrix_element_add: lambda ([[x]], sqrt (apply (&quot;+&quot;, x)))$
(%i2) matrix_element_mult: lambda ([x, y], (x - y)^2)$
(%i3) [a, b, c] . [x, y, z];
                          2          2          2
(%o3)         sqrt((c - z)  + (b - y)  + (a - x) )
(%i4) aa: matrix ([a, b, c], [d, e, f]);
                           [ a  b  c ]
(%o4)                      [         ]
                           [ d  e  f ]
(%i5) bb: matrix ([u, v, w], [x, y, z]);
                           [ u  v  w ]
(%o5)                      [         ]
                           [ x  y  z ]
(%i6) aa . transpose (bb);
               [             2          2          2  ]
               [ sqrt((c - w)  + (b - v)  + (a - u) ) ]
(%o6)  Col 1 = [                                      ]
               [             2          2          2  ]
               [ sqrt((f - w)  + (e - v)  + (d - u) ) ]

                         [             2          2          2  ]
                         [ sqrt((c - z)  + (b - y)  + (a - x) ) ]
                 Col 2 = [                                      ]
                         [             2          2          2  ]
                         [ sqrt((f - z)  + (e - y)  + (d - x) ) ]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>matrix_element_transpose</b>
<a name="IDX804"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p><code>matrix_element_transpose</code> is the operation 
applied to each element of a matrix when it is transposed.
<code>matrix_element_mult</code> can be assigned any unary operator.
The assigned value may be the name of an operator enclosed in quote marks,
the name of a function,
or a lambda expression.
</p>
<p>When <code>matrix_element_transpose</code> equals <code>transpose</code>,
the <code>transpose</code> function is applied to every element.
When <code>matrix_element_transpose</code> equals <code>nonscalars</code>,
the <code>transpose</code> function is applied to every nonscalar element.
If some element is an atom, the <code>nonscalars</code> option applies
<code>transpose</code> only if the atom is declared nonscalar,
while the <code>transpose</code> option always applies <code>transpose</code>.
</p>
<p>The default value, <code>false</code>, means no operation is applied.
</p>
<p>See also <code>matrix_element_add</code> and <code>matrix_element_mult</code>.
</p>
<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) declare (a, nonscalar)$
(%i2) transpose ([a, b]);
                        [ transpose(a) ]
(%o2)                   [              ]
                        [      b       ]
(%i3) matrix_element_transpose: nonscalars$
(%i4) transpose ([a, b]);
                        [ transpose(a) ]
(%o4)                   [              ]
                        [      b       ]
(%i5) matrix_element_transpose: transpose$
(%i6) transpose ([a, b]);
                        [ transpose(a) ]
(%o6)                   [              ]
                        [ transpose(b) ]
(%i7) matrix_element_transpose: lambda ([x], realpart(x) - %i*imagpart(x))$
(%i8) m: matrix ([1 + 5*%i, 3 - 2*%i], [7*%i, 11]);
                     [ 5 %i + 1  3 - 2 %i ]
(%o8)                [                    ]
                     [   7 %i       11    ]
(%i9) transpose (m);
                      [ 1 - 5 %i  - 7 %i ]
(%o9)                 [                  ]
                      [ 2 %i + 3    11   ]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>mattrace</b><i> (<var>M</var>)</i>
<a name="IDX805"></a>
</dt>
<dd><p>Returns the trace (that is, the sum of the elements on the main diagonal) of
the square matrix <var>M</var>.  
</p>
<p><code>mattrace</code> is called by <code>ncharpoly</code>,
an alternative to Maxima's <code>charpoly</code>.
</p>
<p><code>load (&quot;nchrpl&quot;)</code> loads this function.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>minor</b><i> (<var>M</var>, <var>i</var>, <var>j</var>)</i>
<a name="IDX806"></a>
</dt>
<dd><p>Returns the <var>i</var>, <var>j</var> minor of the matrix <var>M</var>.  That is, <var>M</var>
with row <var>i</var> and column <var>j</var> removed.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>ncexpt</b><i> (<var>a</var>, <var>b</var>)</i>
<a name="IDX807"></a>
</dt>
<dd><p>If a non-commutative exponential expression is too
wide to be displayed as <code><var>a</var>^^<var>b</var></code> it appears as <code>ncexpt (<var>a</var>,<var>b</var>)</code>.
</p>
<p><code>ncexpt</code> is not the name of a function or operator;
the name only appears in output, and is not recognized in input.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>ncharpoly</b><i> (<var>M</var>, <var>x</var>)</i>
<a name="IDX808"></a>
</dt>
<dd><p>Returns the characteristic polynomial of the matrix <var>M</var>
with respect to <var>x</var>.  This is an alternative to Maxima's <code>charpoly</code>.
</p>
<p><code>ncharpoly</code> works by computing traces of powers of the given matrix,
which are known to be equal to sums of powers of the roots of the
characteristic polynomial.  From these quantities the symmetric
functions of the roots can be calculated, which are nothing more than
the coefficients of the characteristic polynomial.  <code>charpoly</code> works by
forming the determinant of <code><var>x</var> * ident [n] - a</code>.  Thus <code>ncharpoly</code> wins,
for example, in the case of large dense matrices filled with integers,
since it avoids polynomial arithmetic altogether.
</p>
<p><code>load (&quot;nchrpl&quot;)</code> loads this file.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>newdet</b><i> (<var>M</var>, <var>n</var>)</i>
<a name="IDX809"></a>
</dt>
<dd><p>Computes the determinant of the matrix or array <var>M</var> by the
Johnson-Gentleman tree minor algorithm.
The argument <var>n</var> is the order; it is optional if <var>M</var> is a matrix.
</p>
</dd></dl>

<dl>
<dt><u>Declaration:</u> <b>nonscalar</b>
<a name="IDX810"></a>
</dt>
<dd><p>Makes atoms behave as does a list or matrix with respect to
the dot operator.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>nonscalarp</b><i> (<var>expr</var>)</i>
<a name="IDX811"></a>
</dt>
<dd><p>Returns <code>true</code> if <var>expr</var> is a non-scalar, i.e., it contains
atoms declared as non-scalars, lists, or matrices.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>permanent</b><i> (<var>M</var>, <var>n</var>)</i>
<a name="IDX812"></a>
</dt>
<dd><p>Computes the permanent of the matrix <var>M</var>.  A permanent
is like a determinant but with no sign changes.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>rank</b><i> (<var>M</var>)</i>
<a name="IDX813"></a>
</dt>
<dd><p>Computes the rank of the matrix <var>M</var>.  That is, the order of the
largest non-singular subdeterminant of <var>M</var>.
</p>
<p><var>rank</var> may return the
wrong answer if it cannot determine that a matrix element that is
equivalent to zero is indeed so.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>ratmx</b>
<a name="IDX814"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>ratmx</code> is <code>false</code>, determinant and matrix
addition, subtraction, and multiplication are performed in the
representation of the matrix elements and cause the result of
matrix inversion to be left in general representation.
</p>
<p>When <code>ratmx</code> is <code>true</code>,
the 4 operations mentioned above are performed in CRE form and the
result of matrix inverse is in CRE form.  Note that this may
cause the elements to be expanded (depending on the setting of <code>ratfac</code>)
which might not always be desired.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>row</b><i> (<var>M</var>, <var>i</var>)</i>
<a name="IDX815"></a>
</dt>
<dd><p>Returns the <var>i</var>'th row of the matrix <var>M</var>.
The return value is a matrix.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>scalarmatrixp</b>
<a name="IDX816"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>scalarmatrixp</code> is <code>true</code>, then whenever a 1 x 1 matrix
is produced as a result of computing the dot product of matrices it
is simplified to a scalar, namely the sole element of the matrix.
</p>
<p>When <code>scalarmatrixp</code> is <code>all</code>,
then all 1 x 1 matrices are simplified to scalars.
</p>
<p>When <code>scalarmatrixp</code> is <code>false</code>, 1 x 1 matrices are not simplified to scalars.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>scalefactors</b><i> (<var>coordinatetransform</var>)</i>
<a name="IDX817"></a>
</dt>
<dd><p>Here coordinatetransform
evaluates to the form [[expression1, expression2, ...],
indeterminate1, indeterminat2, ...], where indeterminate1,
indeterminate2, etc. are the curvilinear coordinate variables and
where a set of rectangular Cartesian components is given in terms of
the curvilinear coordinates by [expression1, expression2, ...].
<code>coordinates</code> is set to the vector [indeterminate1, indeterminate2,...],
and <code>dimension</code> is set to the length of this vector.  SF[1], SF[2],
..., SF[DIMENSION] are set to the coordinate scale factors, and <code>sfprod</code>
is set to the product of these scale factors.  Initially, <code>coordinates</code>
is [X, Y, Z], <code>dimension</code> is 3, and SF[1]=SF[2]=SF[3]=SFPROD=1,
corresponding to 3-dimensional rectangular Cartesian coordinates.
To expand an expression into physical components in the current
coordinate system, there is a function with usage of the form
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>setelmx</b><i> (<var>x</var>, <var>i</var>, <var>j</var>, <var>M</var>)</i>
<a name="IDX818"></a>
</dt>
<dd><p>Assigns <var>x</var> to the (<var>i</var>, <var>j</var>)'th element of the matrix <var>M</var>,
and returns the altered matrix.
</p>
<p><code><var>M</var> [<var>i</var>, <var>j</var>]: <var>x</var></code> has the same effect,
but returns <var>x</var> instead of <var>M</var>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>similaritytransform</b><i> (<var>M</var>)</i>
<a name="IDX819"></a>
</dt>
<dt><u>Function:</u> <b>simtran</b><i> (<var>M</var>)</i>
<a name="IDX820"></a>
</dt>
<dd><p><code>similaritytransform</code> computes a similarity transform of the matrix <code>M</code>.
It returns a list which is the output of the
<code>uniteigenvectors</code> command.  In addition if the flag <code>nondiagonalizable</code>
is <code>false</code> two global matrices <code>leftmatrix</code> and <code>rightmatrix</code> are computed.
These matrices have the property that
<code>leftmatrix . <var>M</var> . rightmatrix</code> is a diagonal matrix with the eigenvalues
of <var>M</var> on the diagonal.  If <code>nondiagonalizable</code> is <code>true</code> the left and right
matrices are not computed.
</p>
<p>If the flag <code>hermitianmatrix</code> is <code>true</code>
then <code>leftmatrix</code> is the complex conjugate of the transpose of
<code>rightmatrix</code>.  Otherwise <code>leftmatrix</code> is the inverse of <code>rightmatrix</code>.
</p>
<p><code>rightmatrix</code> is the matrix the columns of which are the unit
eigenvectors of <var>M</var>.  The other flags (see <code>eigenvalues</code> and
<code>eigenvectors</code>) have the same effects since
<code>similaritytransform</code> calls the other functions in the package in order
to be able to form <code>rightmatrix</code>.
</p>
<p><code>load (&quot;eigen&quot;)</code> loads this function.
</p>
<p><code>simtran</code> is a synonym for <code>similaritytransform</code>.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>sparse</b>
<a name="IDX821"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>sparse</code> is <code>true</code>, and if <code>ratmx</code> is <code>true</code>, then <code>determinant</code>
will use special routines for computing sparse determinants.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>submatrix</b><i> (<var>i_1</var>, ..., <var>i_m</var>, <var>M</var>, <var>j_1</var>, ..., <var>j_n</var>)</i>
<a name="IDX822"></a>
</dt>
<dt><u>Function:</u> <b>submatrix</b><i> (<var>i_1</var>, ..., <var>i_m</var>, <var>M</var>)</i>
<a name="IDX823"></a>
</dt>
<dt><u>Function:</u> <b>submatrix</b><i> (<var>M</var>, <var>j_1</var>, ..., <var>j_n</var>)</i>
<a name="IDX824"></a>
</dt>
<dd><p>Returns a new matrix composed of the
matrix <var>M</var> with rows <var>i_1</var>, ..., <var>i_m</var> deleted, and columns <var>j_1</var>, ..., <var>j_n</var> deleted.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>transpose</b><i> (<var>M</var>)</i>
<a name="IDX825"></a>
</dt>
<dd><p>Returns the transpose of <var>M</var>.
</p>
<p>If <var>M</var> is a matrix, the return value is another matrix <var>N</var>
such that <code>N[i,j] = M[j,i]</code>.
</p>
<p>Otherwise <var>M</var> is a list, and the return value is a matrix <var>N</var>
of <code>length (m)</code> rows and 1 column, such that <code>N[i,1] = M[i]</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>triangularize</b><i> (<var>M</var>)</i>
<a name="IDX826"></a>
</dt>
<dd><p>Returns the upper triangular form of the matrix <code>M</code>,
as produced by Gaussian elimination.
The return value is the same as <code>echelon</code>,
except that the leading nonzero coefficient in each row is not normalized to 1.
</p>
<p><code>lu_factor</code> and <code>cholesky</code> are other functions which yield triangularized matrices.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) M: matrix ([3, 7, aa, bb], [-1, 8, 5, 2], [9, 2, 11, 4]);
                       [  3   7  aa  bb ]
                       [                ]
(%o1)                  [ - 1  8  5   2  ]
                       [                ]
                       [  9   2  11  4  ]
(%i2) triangularize (M);
             [ - 1   8         5            2      ]
             [                                     ]
(%o2)        [  0   - 74     - 56         - 22     ]
             [                                     ]
             [  0    0    626 - 74 aa  238 - 74 bb ]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>uniteigenvectors</b><i> (<var>M</var>)</i>
<a name="IDX827"></a>
</dt>
<dt><u>Function:</u> <b>ueivects</b><i> (<var>M</var>)</i>
<a name="IDX828"></a>
</dt>
<dd><p>Computes unit eigenvectors of the matrix <var>M</var>.
The return value is a list of lists, the first sublist of which is the
output of the <code>eigenvalues</code> command, and the other sublists of which are
the unit eigenvectors of the matrix corresponding to those eigenvalues
respectively.
</p>
<p>The flags mentioned in the description of the
<code>eigenvectors</code> command have the same effects in this one as well.
</p>
<p>When <code>knowneigvects</code> is <code>true</code>, the <code>eigen</code> package assumes
that the eigenvectors of the matrix are known to the user and are
stored under the global name <code>listeigvects</code>.  <code>listeigvects</code> should be set
to a list similar to the output of the <code>eigenvectors</code> command.
</p>
<p>If <code>knowneigvects</code> is set to <code>true</code> and the list of eigenvectors is given the
setting of the flag <code>nondiagonalizable</code> may not be correct.  If that is
the case please set it to the correct value.  The author assumes that
the user knows what he is doing and will not try to diagonalize a
matrix the eigenvectors of which do not span the vector space of the
appropriate dimension.
</p>
<p><code>load (&quot;eigen&quot;)</code> loads this function.
</p>
<p><code>ueivects</code> is a synonym for <code>uniteigenvectors</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>unitvector</b><i> (<var>x</var>)</i>
<a name="IDX829"></a>
</dt>
<dt><u>Function:</u> <b>uvect</b><i> (<var>x</var>)</i>
<a name="IDX830"></a>
</dt>
<dd><p>Returns <em><var>x</var>/norm(<var>x</var>)</em>;
this is a unit vector in the same direction as <var>x</var>.
</p>
<p><code>load (&quot;eigen&quot;)</code> loads this function.
</p>
<p><code>uvect</code> is a synonym for <code>unitvector</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>vectorsimp</b><i> (<var>expr</var>)</i>
<a name="IDX831"></a>
</dt>
<dd><p>Applies simplifications and expansions according
to the following global flags:
</p>
<p><code>expandall</code>, <code>expanddot</code>, <code>expanddotplus</code>, <code>expandcross</code>, <code>expandcrossplus</code>,
<code>expandcrosscross</code>, <code>expandgrad</code>, <code>expandgradplus</code>, <code>expandgradprod</code>,
<code>expanddiv</code>, <code>expanddivplus</code>, <code>expanddivprod</code>, <code>expandcurl</code>, <code>expandcurlplus</code>,
<code>expandcurlcurl</code>, <code>expandlaplacian</code>, <code>expandlaplacianplus</code>,
and <code>expandlaplacianprod</code>.
</p>
<p>All these flags have default value <code>false</code>. The <code>plus</code> suffix refers to
employing additivity or distributivity.  The <code>prod</code> suffix refers to the
expansion for an operand that is any kind of product.
</p>
<dl compact="compact">
<dt> <code>expandcrosscross</code></dt>
<dd><p>Simplifies <em>p ~ (q ~ r)</em> to <em>(p . r)*q - (p . q)*r</em>.
</p></dd>
<dt> <code>expandcurlcurl</code></dt>
<dd><p>Simplifies <em>curl curl p</em> to <em>grad div p + div grad p</em>.
</p></dd>
<dt> <code>expandlaplaciantodivgrad</code></dt>
<dd><p>Simplifies <em>laplacian p</em> to <em>div grad p</em>.
</p></dd>
<dt> <code>expandcross</code></dt>
<dd><p>Enables <code>expandcrossplus</code> and <code>expandcrosscross</code>.
</p></dd>
<dt> <code>expandplus</code></dt>
<dd><p>Enables <code>expanddotplus</code>, <code>expandcrossplus</code>, <code>expandgradplus</code>,
<code>expanddivplus</code>, <code>expandcurlplus</code>, and <code>expandlaplacianplus</code>.
</p></dd>
<dt> <code>expandprod</code></dt>
<dd><p>Enables <code>expandgradprod</code>, <code>expanddivprod</code>, and <code>expandlaplacianprod</code>.
</p></dd>
</dl>

<p>These flags have all been declared <code>evflag</code>.
</p>

</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>vect_cross</b>
<a name="IDX832"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>vect_cross</code> is <code>true</code>, it allows DIFF(X~Y,T) to work where
~ is defined in SHARE;VECT (where VECT_CROSS is set to <code>true</code>, anyway.)
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>zeromatrix</b><i> (<var>m</var>, <var>n</var>)</i>
<a name="IDX833"></a>
</dt>
<dd><p>Returns an <var>m</var> by <var>n</var> matrix, all elements of which are zero.
</p>
</dd></dl>

<dl>
<dt><u>Special symbol:</u> <b>[</b>
<a name="IDX834"></a>
</dt>
<dt><u>Special symbol:</u> <b>]</b>
<a name="IDX835"></a>
</dt>
<dd><p><code>[</code> and <code>]</code> mark the beginning and end, respectively, of a list.
</p>
<p><code>[</code> and <code>]</code> also enclose the subscripts of
a list, array, hash array, or array function.
</p>
<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) x: [a, b, c];
(%o1)                       [a, b, c]
(%i2) x[3];
(%o2)                           c
(%i3) array (y, fixnum, 3);
(%o3)                           y
(%i4) y[2]: %pi;
(%o4)                          %pi
(%i5) y[2];
(%o5)                          %pi
(%i6) z['foo]: 'bar;
(%o6)                          bar
(%i7) z['foo];
(%o7)                          bar
(%i8) g[k] := 1/(k^2+1);
                                  1
(%o8)                     g  := ------
                           k     2
                                k  + 1
(%i9) g[10];
                                1
(%o9)                          ---
                               101
</pre></td></tr></table>
</dd></dl>

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