Matrices are an important tool in algebra. A matrix nicely represents a homomorphism between two vector spaces with respect to a choice of bases for the vector spaces. Also matrices represent systems of linear equations.
In GAP3 matrices are represented by list of vectors (see Vectors). The vectors must all have the same length, and their elements must lie in a common field. The field may be the field of rationals (see Rationals), a cyclotomic field (see Cyclotomics), a finite field (see Finite Fields), or a library and/or user defined field (or ring) such as a polynomial ring (see Polynomials).
The first section in this chapter describes the operations applicable to matrices (see Operations for Matrices). The next sections describes the function that tests whether an object is a matrix (see IsMat). The next sections describe the functions that create certain matrices (see IdentityMat, NullMat, TransposedMat, and KroneckerProduct). The next sections describe functions that compute certain characteristic values of matrices (see DimensionsMat, TraceMat, DeterminantMat, RankMat, and OrderMat). The next sections describe the functions that are related to the interpretation of a matrix as a system of linear equations (see TriangulizeMat, BaseMat, NullspaceMat, and SolutionMat). The last two sections describe the functions that diagonalize an integer matrix (see DiagonalizeMat and ElementaryDivisorsMat).
Because matrices are just a special case of lists, all operations and functions for lists are applicable to matrices also (see chapter Lists). This especially includes accessing elements of a matrix (see List Elements), changing elements of a matrix (see List Assignment), and comparing matrices (see Comparisons of Lists).
mat + scalar
scalar + mat
This forms evaluates to the sum of the matrix mat and the scalar scalar. The elements of mat and scalar must lie in a common field. The sum is a new matrix where each entry is the sum of the corresponding entry of mat and scalar.
mat1 + mat2
This form evaluates to the sum of the two matrices mat1 and mat2, which must have the same dimensions and whose elements must lie in a common field. The sum is a new matrix where each entry is the sum of the corresponding entries of mat1 and mat2.
mat - scalar
scalar - mat
mat1 - mat2
The definition for the - operator are similar to the above definitions
for the + operator, except that - subtracts of course.
mat * scalar
scalar * mat
This forms evaluate to the product of the matrix mat and the scalar scalar. The elements of mat and scalar must lie in a common field. The product is a new matrix where each entry is the product of the corresponding entries of mat and scalar.
vec * mat
This form evaluates to the product of the vector vec and the matrix
mat. The length of vec and the number of rows of mat must be
equal. The elements of vec and mat must lie in a common field. If
vec is a vector of length n and mat is a matrix with n rows and
m columns, the product is a new vector of length m. The element at
position i is the sum of vec[l] * mat[l][i] with l
running from 1 to n.
mat * vec
This form evaluates to the product of the matrix mat and the vector
vec. The number of columns of mat and the length of vec must be
equal. The elements of mat and vec must lie in a common field. If
mat is a matrix with m rows and n columns and vec is a vector of
length n, the product is a new vector of length m. The element at
position i is the sum of mat[i][l] * vec[l] with l
running from 1 to n.
mat1 * mat2
This form evaluates to the product of the two matrices mat1 and mat2.
The number of columns of mat1 and the number of rows of mat2 must be
equal. The elements of mat1 and mat2 must lie in a common field. If
mat1 is a matrix with m rows and n columns and mat2 is a matrix
with n rows and o columns, the result is a new matrix with m rows
and o columns. The element in row i at position k of the product
is the sum of mat1[i][l] * mat2[l][k] with l running
from 1 to n.
mat1 / mat2
scalar / mat
mat / scalar
vec / mat
In general left / right is defined as left * right^-1.
Thus in the above forms the right operand must always be invertable.
mat ^ int
This form evaluates to the int-th power of the matrix mat. mat must be a square matrix, int must be an integer. If int is negative, mat must be invertible. If int is 0, the result is the identity matrix, even if mat is not invertible.
mat1 ^ mat2
This form evaluates to the conjugation of the matrix mat1 by the matrix
mat2, i.e., to mat2^-1 * mat1 * mat2. mat2 must be
invertible and mat1 must be such that these product can be computed.
vec ^ mat
This is in every respect equivalent to vec * mat. This
operations reflects the fact that matrices operate on the vector space by
multiplication from the right.
scalar + matlist
matlist + scalar
scalar - matlist
matlist - scalar
scalar * matlist
matlist * scalar
matlist / scalar
A scalar scalar may also be added, subtracted, multiplied with, or divide into a whole list of matrices matlist. The result is a new list of matrices where each matrix is the result of performing the operation with the corresponding matrix in matlist.
mat * matlist
matlist * mat
A matrix mat may also be multiplied with a whole list of matrices matlist. The result is a new list of matrices, where each matrix is the product of mat and the corresponding matrix in matlist.
matlist / mat
This form evaluates to matlist * mat^-1. mat must of course
be invertable.
vec * matlist
This form evaluates to the product of the vector vec and the list of
matrices mat. The length l of vec and matlist must be equal.
All matrices in matlist must have the same dimensions. The elements of
vec and the elements of the matrices in matlist must lie in a common
field. The product is the sum of vec[i] * matlist[i] with
i running from 1 to l.
Comm( mat1, mat2 )
Comm returns the commutator of the matrices mat1 and mat2, i.e.,
mat1^-1 * mat2^-1 * mat1 * mat2. mat1 and mat2
must be invertable and such that these product can be computed.
There is one exception to the rule that the operands or their elements
must lie in common field. It is allowed that one operand is a finite
field element, a finite field vector, a finite field matrix, or a list of
finite field matrices, and the other operand is an integer, an integer
vector, an integer matrix, or a list of integer matrices. In this case
the integers are interpreted as int * GF.one, where GF is the
finite field (see Operations for Finite Field Elements).
For all the above operations the result is new, i.e., not identical to any other list (see Identical Lists). This is the case even if the result is equal to one of the operands, e.g., if you add zero to a matrix.
IsMat( obj )
IsMat return true if obj, which can be an object of arbitrary type,
is a matrix and false otherwise. Will cause an error if obj is an
unbound variable.
gap> IsMat( [ [ 1, 0 ], [ 0, 1 ] ] );
true # a matrix is a list of vectors
gap> IsMat( [ [ 1, 2, 3, 4, 5 ] ] );
true
gap> IsMat( [ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] ] );
true
gap> IsMat( [ [ Z(2)^0, 0 ], [ 0, Z(2)^0 ] ] );
false # Z(2)^0 and 0 do not lie in a common field
gap> IsMat( [ 1, 0 ] );
false # a vector is not a matrix
gap> IsMat( 1 );
false # neither is a scalar
IdentityMat( n )
IdentityMat( n, F )
IdentityMat returns the identity matrix with n rows and n columns
over the field F. If no field is given, IdentityMat returns the
identity matrix over the field of rationals. Each call to IdentityMat
returns a new matrix, so it is safe to modify the result.
gap> IdentityMat( 3 );
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
gap> PrintArray( last );
[ [ 1, 0, 0 ],
[ 0, 1, 0 ],
[ 0, 0, 1 ] ]
gap> PrintArray( IdentityMat( 3, GF(2) ) );
[ [ Z(2)^0, 0*Z(2), 0*Z(2) ],
[ 0*Z(2), Z(2)^0, 0*Z(2) ],
[ 0*Z(2), 0*Z(2), Z(2)^0 ] ]
NullMat( m )
NullMat( m, n )
NullMat( m, n, F )
NullMat returns the null matrix with m rows and n columns over the
field F; if n is omitted, it is assumed equal to m. If no field is
given, NullMat returns the null matrix over the field of rationals. Each
call to NullMat returns a new matrix, so it is safe to modify the result.
gap> PrintArray( NullMat( 2, 3 ) );
[ [ 0, 0, 0 ],
[ 0, 0, 0 ] ]
gap> PrintArray( NullMat( 2, 2, GF(2) ) );
[ [ 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2) ] ]
TransposedMat( mat )
TransposedMat returns the transposed of the matrix mat. The
transposed matrix is a new matrix trn, such that trn[i][k] is
mat[k][i].
gap> TransposedMat( [ [ 1, 2 ], [ 3, 4 ] ] );
[ [ 1, 3 ], [ 2, 4 ] ]
gap> TransposedMat( [ [ 1..5 ] ] );
[ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ] ]
KroneckerProduct( mat1, ..., matn )
KroneckerProduct returns the Kronecker product of the matrices mat1,
..., matn. If mat1 is a m by n matrix and mat2 is a o by p
matrix, the Kronecker product of mat1 by mat2 is a m*o by
n*p matrix, such that the entry in row (i1-1)*o+i2 at
position (k1-1)*p+k2 is mat1[i1][k1] *
mat2[i2][k2].
gap> mat1 := [ [ 0, -1, 1 ], [ -2, 0, -2 ] ];;
gap> mat2 := [ [ 1, 1 ], [ 0, 1 ] ];;
gap> PrintArray( KroneckerProduct( mat1, mat2 ) );
[ [ 0, 0, -1, -1, 1, 1 ],
[ 0, 0, 0, -1, 0, 1 ],
[ -2, -2, 0, 0, -2, -2 ],
[ 0, -2, 0, 0, 0, -2 ] ]
DimensionsMat( mat )
DimensionsMat returns the dimensions of the matrix mat as a list of
two integers. The first entry is the number of rows of mat, the second
entry is the number of columns.
gap> DimensionsMat( [ [ 1, 2, 3 ], [ 4, 5, 6 ] ] );
[ 2, 3 ]
gap> DimensionsMat( [ [ 1 .. 5 ] ] );
[ 1, 5 ]
IsDiagonalMat( mat )
mat must be a matrix. This function returns true if mat is square and
all entries mat[i][j] with i<>j are equal to 0*mat[i][j] and
false otherwise.
gap> mat := [ [ 1, 2 ], [ 3, 1 ] ];;
gap> IsDiagonalMat( mat );
false
IsLowerTriangularMat( mat )
mat must be a matrix. This function returns true if all entries
mat[i][j] with j>i are equal to 0*mat[i][j] and false
otherwise.
gap> a := [ [ 1, 2 ], [ 3, 1 ] ];;
gap> IsLowerTriangularMat( a );
false
gap> a[1][2] := 0;;
gap> IsLowerTriangularMat( a );
true
IsUpperTriangularMat( mat )
mat must be a matrix. This function returns true if all entries
mat[i][j] with j < i are equal to 0*mat[i][j] and false
otherwise.
gap> a := [ [ 1, 2 ], [ 3, 1 ] ];;
gap> IsUpperTriangularMat( a );
false
gap> a[2][1] := 0;;
gap> IsUpperTriangularMat( a );
true
DiagonalOfMat( mat )
This function returns the list of diagonal entries of the matrix mat,
that is the list of mat[i][i].
gap> mat := [ [ 1, 2 ], [ 3, 1 ] ];;
gap> DiagonalOfMat( mat );
[ 1, 1 ]
DiagonalMat( mat1, ... , matn )
returns the block diagonal direct sum of the matrices mat1, ..., matn. Blocks of size 1× 1 may be given as scalars.
gap> C1 := [ [ 2, -1, 0, 0 ],
> [ -1, 2, -1, 0 ],
> [ 0, -1, 2, -1 ],
> [ 0, 0, -1, 2 ] ];;
gap> C2 := [ [ 2, 0, -1 ],
> [ 0, 2, -1 ],
> [ -1, -1, 2 ] ];;
gap> PrintArray( DiagonalMat( C1, 3, C2 ) );
[ [ 2, -1, 0, 0, 0, 0, 0, 0 ],
[ -1, 2, -1, 0, 0, 0, 0, 0 ],
[ 0, -1, 2, -1, 0, 0, 0, 0 ],
[ 0, 0, -1, 2, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 3, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 2, 0, -1 ],
[ 0, 0, 0, 0, 0, 0, 2, -1 ],
[ 0, 0, 0, 0, 0, -1, -1, 2 ] ]
To make a diagonal matrix with a specified diagonal d use
ApplyFunc(DiagonalMat, d ).
PermutationMat( perm, dim[, F] ) ( function )
returns a matrix in dimension dim over the field given by F (i.e. the
smallest field containing the element F or F itself if it is a field) that
represents the permutation perm acting by permuting the basis vectors as it
permutes points.
PermutationMat( perm, dim [,F])
returns a matrix in dimension dim over the field F (by default the rationals) that represents the permutation perm acting by permuting the basis vectors as it permutes points.
gap> PermutationMat((1,2,3),4);
[ [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 1 ] ]
TraceMat( mat )
TraceMat returns the trace of the square matrix mat. The trace is
the sum of all entries on the diagonal of mat.
gap> TraceMat( [ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ] );
15
gap> TraceMat( IdentityMat( 4, GF(2) ) );
0*Z(2)
DeterminantMat( mat )
DeterminantMat returns the determinant of the square matrix mat. The
determinant is defined by
∑p ∈ Symm(n){sign(p)∏i=1nmat[i][ip]}.
gap> DeterminantMat( [ [ 1, 2 ], [ 3, 4 ] ] );
-2
gap> DeterminantMat( [ [ 0*Z(3), Z(3)^0 ], [ Z(3)^0, Z(3) ] ] );
Z(3)
Note that DeterminantMat does not use the above definition to compute
the result. Instead it performs a Gaussian elimination. For large
rational matrices this may take very long, because the entries may become
very large, even if the final result is a small integer.
RankMat( mat )
RankMat returns the rank of the matrix mat. The rank is defined as
the dimension of the vector space spanned by the rows of mat. It
follows that a n by n matrix is invertible exactly if its rank is
n.
gap> RankMat( [ [ 4, 1, 2 ], [ 3, -1, 4 ], [ -1, -2, 2 ] ] );
2
Note that RankMat performs a Gaussian elimination. For large rational
matrices this may take very long, because the entries may become very
large.
OrderMat( mat )
OrderMat returns the order of the invertible square matrix mat. The
order ord is the smallest positive integer such that mat^ord is
the identity.
gap> OrderMat( [ [ 0*Z(2), 0*Z(2), Z(2)^0 ],
> [ Z(2)^0, Z(2)^0, 0*Z(2) ],
> [ Z(2)^0, 0*Z(2), 0*Z(2) ] ] );
4
OrderMat first computes ord1 such that the first standard basis
vector is mapped by mat^ord1 onto itself. It does this by
applying mat repeatedly to the first standard basis vector. Then it
computes mat1 as mat1^ord1. Then it computes ord2 such that
the second standard basis vector is mapped by mat1^ord2 onto
itself. This process is repeated until all basis vectors are mapped onto
themselves. OrderMat warns you that the order may be infinite, when it
finds that the order must be larger than 1000.
TriangulizeMat( mat )
TriangulizeMat brings the matrix mat into upper triangular form. Note
that mat is changed. A matrix is in upper triangular form when the first
nonzero entry in each row is one and lies further to the right than the
first nonzero entry in the previous row. Furthermore, above the first
nonzero entry in each row all entries are zero. Note that the matrix will
have trailing zero rows if the rank of mat is not maximal. The rows of
the resulting matrix span the same vectorspace than the rows of the
original matrix mat. The function returns the indices of the lines of the
orginal matrix corresponding to the non-zero lines of the triangulized
matrix.
gap> m := [ [ 0, -3, -1 ], [ -3, 0, -1 ], [ 2, -2, 0 ] ];;
gap> TriangulizeMat( m ); m;
[ 2, 1 ]
[ [ 1, 0, 1/3 ], [ 0, 1, 1/3 ], [ 0, 0, 0 ] ]
Note that for large rational matrices TriangulizeMat may take very long,
because the entries may become very large during the Gaussian elimination,
even if the final result contains only small integers.
BaseMat( mat )
BaseMat returns a standard base for the vector space spanned by the
rows of the matrix mat. The standard base is in upper triangular form.
That means that the first nonzero vector in each row is one and lies
further to the right than the first nonzero entry in the previous row.
Furthermore, above the first nonzero entry in each row all entries are
zero.
gap> BaseMat( [ [ 0, -3, -1 ], [ -3, 0, -1 ], [ 2, -2, 0 ] ] );
[ [ 1, 0, 1/3 ], [ 0, 1, 1/3 ] ]
Note that for large rational matrices BaseMat may take very long,
because the entries may become very large during the Gaussian
elimination, even if the final result contains only small integers.
NullspaceMat( mat )
NullspaceMat returns a base for the nullspace of the matrix mat. The
nullspace is the set of vectors vec such that vec * mat is the
zero vector. The returned base is the standard base for the nullspace
(see BaseMat).
gap> NullspaceMat( [ [ 2, -4, 1 ], [ 0, 0, -4 ], [ 1, -2, -1 ] ] );
[ [ 1, 3/4, -2 ] ]
Note that for large rational matrices NullspaceMat may take very long,
because the entries may become very large during the Gaussian
elimination, even if the final result only contains small integers.
SolutionMat( mat, vec )
SolutionMat returns one solution of the equation x * mat =
vec or false if no such solution exists.
gap> SolutionMat( [ [ 2, -4, 1 ], [ 0, 0, -4 ], [ 1, -2, -1 ] ],
> [ 10, -20, -10 ] );
[ 5, 15/4, 0 ]
gap> SolutionMat( [ [ 2, -4, 1 ], [ 0, 0, -4 ], [ 1, -2, -1 ] ],
> [ 10, 20, -10 ] );
false
Note that for large rational matrices SolutionMat may take very long,
because the entries may become very large during the Gaussian
elimination, even if the final result only contains small integers.
DiagonalizeMat( mat )
DiagonalizeMat transforms the integer matrix mat by multiplication
with unimodular (i.e., determinant +1 or -1) integer matrices from the
left and from the right into diagonal form (i.e., only diagonal entries
are nonzero). Note that DiagonalizeMat changes mat and returns
nothing. If there are several diagonal matrices to which mat is
equivalent, it is not specified which one is computed, except that all
zero entries on the diagonal are collected at the lower right end (see
ElementaryDivisorsMat).
gap> m := [ [ 0, -1, 1 ], [ -2, 0, -2 ], [ 2, -2, 4 ] ];;
gap> DiagonalizeMat( m ); m;
[ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 0 ] ]
Note that for large integer matrices DiagonalizeMat may take very long,
because the entries may become very large during the computation, even if
the final result only contains small integers.
ElementaryDivisorsMat( mat )
ElementaryDivisors returns a list of the elementary divisors, i.e., the
unique d with d[i] divides d[i+1] and mat is equivalent
to a diagonal matrix with the elements d[i] on the diagonal (see
DiagonalizeMat).
gap> m := [ [ 0, -1, 1 ], [ -2, 0, -2 ], [ 2, -2, 4 ] ];;
gap> ElementaryDivisorsMat( m );
[ 1, 2, 0 ]
PrintArray( mat )
PrintArray displays the matrix mat in a pretty way.
gap> m := [[1,2,3,4],[5,6,7,8],[9,10,11,12]];
[ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8 ], [ 9, 10, 11, 12 ] ]
gap> PrintArray( m );
[ [ 1, 2, 3, 4 ],
[ 5, 6, 7, 8 ],
[ 9, 10, 11, 12 ] ]
gap3-jm