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\section{Matrix Operations}

\begin{Operator}{COFACTOR}
\index{matrix}
The operator \name{cofactor} returns the cofactor of the element in row
\meta{row} and column \meta{column} of a \nameref{matrix}.  Errors occur
if \meta{row} or \meta{column} do not evaluate to integer expressions or if
the matrix is not square.

\begin{Syntax}
\name{cofactor}\(\meta{matrix\_expression},\meta{row},\meta{column}\)
% COFACTOR(EXPRN:matrix_expression,ROW:integer,COLUMN:integer):algebraic
\end{Syntax}

\begin{Examples}
cofactor(mat((a,b,c),(d,e,f),(p,q,r)),2,2); & A*R - C*P \\
cofactor(mat((a,b,c),(d,e,f)),1,1); &  ***** non-square matrix
\end{Examples}
\end{Operator}


\begin{Operator}{DET}
\index{matrix}\index{determinant}
The \name{det} operator returns the determinant of its 
(square \nameref{matrix}) argument.

\begin{Syntax}
\name{det}\(\meta{expression}\) or \name{det} \meta{expression}
\end{Syntax}

\meta{expression} must evaluate to a square matrix.

\begin{Examples}

matrix m,n; \\

m := mat((a,b),(c,d));      &  \begin{multilineoutput}{4cm}
M(1,1) := A
M(1,2) := B
M(2,1) := C
M(2,2) := D
                               \end{multilineoutput}\\
det m;                      &      A*D - B*C \\
n := mat((1,2),(1,2));      &  \begin{multilineoutput}{4cm}
N(1,1) := 1
N(1,2) := 2
N(2,1) := 1
N(2,2) := 2
                               \end{multilineoutput} \\

det(n);                     &      0 \\

det(5);                     &      5
\end{Examples}

\begin{Comments}
Given a numerical argument, \name{det} returns the number.  However, given a
variable name that has not been declared of type matrix, or a non-square
matrix, \name{det} returns an error message.
\end{Comments}
\end{Operator}


\begin{Operator}{MAT}
\index{matrix}
The \name{mat} operator is used to represent a two-dimensional 
\nameref{matrix}.
\begin{Syntax}
\name{mat}\(\(\meta{expr}\{,\meta{expr}\}\optional\)%
            \{\(\meta{expr}\{\name{,}\meta{expr}\}\optional\)\}\optional\)
\end{Syntax}


\meta{expr} may be any valid REDUCE scalar expression.

\begin{Examples}
mat((1,2),(3,4));            & 
\begin{multilineoutput}{6cm}
MAT(1,1) := 1
MAT(2,3) := 2
MAT(2,1) := 3
MAT(2,2) := 4
\end{multilineoutput}\\
mat(2,1);                    & \begin{multilineoutput}{6cm}
***** Matrix mismatch
Cont? (Y or N) 
\end{multilineoutput}\\
matrix qt;  \\
qt := ws;                    & \begin{multilineoutput}{6cm}
QT(1,1) := 1
QT(1,2) := 2
QT(2,1) := 3
QT(2,2) := 4 
\end{multilineoutput}\\
matrix a,b; \\
a := mat((x),(y),(z));       & \begin{multilineoutput}{6cm}
A(1,1) := X
A(2,1) := Y
A(3,1) := Z 
\end{multilineoutput}\\
b := mat((sin x,cos x,1));   & \begin{multilineoutput}{6cm}
B(1,1) := SIN(X)
B(1,2) := COS(X)
B(1,3) := 1
\end{multilineoutput}
\end{Examples}

\begin{Comments}
Matrices need not have a size declared (unlike arrays).  \name{mat}
redimensions a matrix variable as needed.  It is necessary, of course,
that all rows be the same length.  An anonymous matrix, as shown in the
first example, must be named before it can be referenced (note error
message).  When using \name{mat} to fill a \IFTEX{$1 \times n$}{1 x n}
matrix, the row of values must be inside a second set of parentheses, to
eliminate ambiguity.
\end{Comments}
\end{Operator}


\begin{Operator}{MATEIGEN}
\index{matrix}\index{eigenvalue}
The \name{mateigen} operator calculates the eigenvalue equation and the
corresponding eigenvectors of a \nameref{matrix}.
\begin{Syntax}
\name{mateigen}\(\meta{matrix-id},\meta{tag-id}\)
\end{Syntax}


\meta{matrix-id} must be a declared matrix of values, and \meta{tag-id} must be 
a legal REDUCE identifier.

\begin{Examples}
aa := mat((2,5),(1,0))\$ \\
mateigen(aa,alpha);          & 
\begin{multilineoutput}{2cm}
\{\{ALPHA^{2} - 2*ALPHA - 5,
  1,
  MAT(1,1) := \rfrac{5*ARBCOMPLEX(1)}{ALPHA - 2},

  MAT(2,1) := ARBCOMPLEX(1)
  \}\}
\end{multilineoutput}\\
charpoly := first first ws;  &       CHARPOLY := ALPHA^{2} - 2*ALPHA - 5 \\
bb := mat((1,0,1),(1,1,0),(0,0,1))\$ \\
mateigen(bb,lamb);          &  
\begin{multilineoutput}{6cm}
\{\{LAMB - 1,3,
  [      0      ]
  [ARBCOMPLEX(2)]
  [      0      ]
  \}\}
\end{multilineoutput}
\end{Examples}


\begin{Comments}
The \name{mateigen} operator returns a list of lists of three
elements.  The first element is a square free factor of the characteristic
polynomial; the second element is its multiplicity; and the third element
is the corresponding eigenvector.  If the characteristic polynomial can be
completely factored, the product of the first elements of all the sublists
will produce the minimal polynomial.  You can access the various parts of
the answer with the usual list access operators.

If the matrix is degenerate, more than one eigenvector can be produced for
the same eigenvalue, as shown by more than one arbitrary variable in the
eigenvector.  The identification numbers of the arbitrary complex variables
shown in the examples above may not be the same as yours.  Note that since
\name{lambda} is a reserved word in REDUCE, you cannot use it as a {\it
tag-id} for this operator.
\end{Comments}
\end{Operator}


\begin{Declaration}{MATRIX}
Identifiers are declared to be of type \name{matrix}.
\begin{Syntax}
\name{matrix} \meta{identifier} \&option \(\meta{index},\meta{index}\)\\
          \{,\meta{identifier} \&option
          \(\meta{index},\meta{index}\)\}\optional
\end{Syntax}

\meta{identifier} must not be an already-defined operator or array or 
the name of a scalar variable.  Dimensions are optional, and if used appear 
inside parentheses.  \meta{index} must be a positive integer. 

\begin{Examples}
matrix a,b(1,4),c(4,4); \\
b(1,1);                      &          0 \\
a(1,1);                      &          ***** Matrix A not set \\
a := mat((x0,y0),(x1,y1));   & \begin{multilineoutput}{6cm}
A(1,1) := X0
A(1,2) := Y0
A(2,1) := X0
A(2,2) := X1
\end{multilineoutput}\\
length a;                    &          \{2,2\} \\
b := a**2;                   & \begin{multilineoutput}{6cm}
B(1,1) := X0^{2} + X1*Y0
B(1,2) := Y0*(X0 + Y1)
B(2,1) := X1*(X0 + Y1)
B(2,2) := X1*Y0 + Y1^{2}
\end{multilineoutput}
\end{Examples}

\begin{Comments}
When a matrix variable has not been dimensioned, matrix elements cannot be
referenced until the matrix is set by the \nameref{mat} operator.  When a
matrix is dimensioned in its declaration, matrix elements are set to 0.
Matrix elements cannot stand for themselves.  When you use \nameref{let} on
a matrix element, there is no effect unless the element contains a
constant, in which case an error message is returned.  The same behavior
occurs with \nameref{clear}.  Do \meta{not} use \nameref{clear} to try to
set a matrix element to 0. \nameref{let} statements can be applied to
matrices as a whole, if the right-hand side of the expression is a matrix
expression, and the left-hand side identifier has been declared to be a matrix.

Arithmetical operators apply to matrices of the correct dimensions.  The
operators \name{+} and \name{-} can be used with matrices of the same
dimensions.  The operator \name{*} can be used to multiply
\IFTEX{$m \times n$}{m x n} matrices by \IFTEX{$n \times p$}{n x p}
matrices.  Matrix multiplication is non-commutative.  Scalars can also be
multiplied with matrices, with the result that each element of the matrix
is multiplied by the scalar.  The operator \name{/} applied to two
matrices computes the first matrix multiplied by the inverse of the
second, if the inverse exists, and produces an error message otherwise.
Matrices can be divided by scalars, which results in dividing each element
of the matrix.  Scalars can also be divided by matrices when the matrices
are invertible, and the result is the multiplication of the scalar by the
inverse of the matrix.  Matrix inverses can by found by \name{1/A} or
\name{/A}, where \name{A} is a matrix.  Square matrices can be raised to
positive integer powers, and also to negative integer powers if they are
nonsingular.

When a matrix variable is assigned to the results of a calculation, the
matrix is redimensioned if necessary.
\end{Comments}
\end{Declaration}


\begin{Operator}{NULLSPACE}
\index{matrix}
\begin{Syntax}
\name{nullspace}(\meta{matrix\_expression})
\end{Syntax}

\meta{nullspace} calculates for its \nameref{matrix} argument, 
\name{a}, a list of
linear independent vectors (a basis) whose linear combinations satisfy the
equation $a x = 0$.  The basis is provided in a form such that as many
upper components as possible are isolated.

\begin{Examples}
nullspace mat((1,2,3,4),(5,6,7,8)); &
\begin{multilineoutput}{6cm}
       \{
         [ 1  ]
         [    ]
         [ 0  ]
         [    ]
         [ - 3]
         [    ]
         [ 2  ]
         ,
         [ 0  ]
         [    ]
         [ 1  ]
         [    ]
         [ - 2]
         [    ]
         [ 1  ]
        \}
\end{multilineoutput}
\end{Examples}

\begin{Comments}
Note that with \name{b := nullspace a}, the expression \name{length b} is
the {\em nullity\/} of A, and that \name{second length a - length b}
calculates the {\em rank\/} of A.  The rank of a matrix expression can
also be found more directly by the \nameref{rank} operator.

In addition to the REDUCE matrix form, \name{nullspace} accepts as input a
matrix given as a \nameref{list} of lists, that is interpreted as a row matrix.  If
that form of input is chosen, the vectors in the result will be
represented by lists as well.  This additional input syntax facilitates
the use of \name{nullspace} in applications different from classical linear
algebra.
\end{Comments}

\end{Operator}


\begin{Operator}{RANK}
\index{matrix}
\begin{Syntax}
\name{rank}(\meta{matrix\_expression})
\end{Syntax}
\name{rank} calculates the rank of its matrix argument.

\begin{Examples}
rank mat((a,b,c),(d,e,f));  & 2
\end{Examples}

\begin{Comments}
The argument to \name{rank} can also be a \nameref{list} of lists, interpreted
either as a row matrix or a set of equations.  If that form of input is
chosen, the vectors in the result will be represented by lists as well.
This additional input syntax facilitates the use of \name{rank} in
applications different from classical linear algebra.
\end{Comments}

\end{Operator}


\begin{Operator}{TP}
\index{transpose}\index{matrix}
The \name{tp} operator returns the transpose of its \nameref{matrix}
 argument.
\begin{Syntax}
\name{tp} \meta{identifier} or  \name{tp}\(\meta{identifier}\)
\end{Syntax}

\meta{identifier} must be a matrix, which either has had its dimensions set
in its declaration, or has had values put into it by \name{mat}.

\begin{Examples}
matrix m,n; \\
m := mat((1,2,3),(4,5,6))$ \\
n := tp m;                   &
\begin{multilineoutput}{6cm}
N(1,1) := 1
N(1,2) := 4
N(2,1) := 2
N(2,2) := 5
N(3,1) := 3
N(3,2) := 6
\end{multilineoutput}
\end{Examples}
\begin{Comments}
In an assignment statement involving \name{tp}, the matrix identifier on the
left-hand side is redimensioned to the correct size for the transpose.
\end{Comments}
\end{Operator}


\begin{Operator}{TRACE}
\index{matrix}
The \name{trace} operator finds the trace of its \nameref{matrix} argument.
\begin{Syntax}
\name{trace}\(\meta{expression}\) or \name{trace} \meta{simple\_expression}
\end{Syntax}

\meta{expression} or \meta{simple\_expression} must evaluate to a square
matrix.

\begin{Examples}
matrix a; \\
a := mat((x1,y1),(x2,y2))\$ \\
trace a;                     &         X1 + Y2
\end{Examples}
\begin{Comments}
The trace is the sum of the entries along the diagonal of a square matrix.
Given a non-matrix expression, or a non-square matrix, \name{trace} returns
an error message.
\end{Comments}
\end{Operator}
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