Matrices are represented in GAP by lists of row vectors (see Row Vectors). The vectors must all have the same length, and their elements must lie in a common ring.
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).
IsMatrix(
obj ) C
A matrix is a list of lists of equal length whose entries lie in a common ring.
Note that matrices may have different multiplications,
besides the usual matrix product there is for example the Lie product.
So there are categories such as IsOrdinaryMatrix
and IsLieMatrix
(see IsOrdinaryMatrix, IsLieMatrix)
that describe the matrix multiplication.
One can form the product of two matrices only if they support the same
multiplication.
gap> mat:=[[1,2,3],[4,5,6],[7,8,9]]; [ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ] gap> IsMatrix(mat); true
Note also the filter IsTable
(see section IsTable)
which may be more appropriate than IsMatrix
for some purposes.
Note that the empty list '[ ]' and more complex ``empty'' structures
such as [[ ]]
are not matrices, although special
methods allow them be used in place of matrices in some
situations. See EmptyMatrix below.
gap> [[0]]*[[]]; [ [ ] ] gap> IsMatrix([[]]); false
IsOrdinaryMatrix(
obj ) C
An ordinary matrix is a matrix whose multiplication is the ordinary matrix multiplication.
Each matrix in internal representation is in the category
IsOrdinaryMatrix
,
and arithmetic operations with objects in IsOrdinaryMatrix
produce
again matrices in IsOrdinaryMatrix
.
Note that we want that Lie matrices shall be matrices that behave in the same way as ordinary matrices, except that they have a different multiplication. So we must distinguish the different matrix multiplications, in order to be able to describe the applicability of multiplication, and also in order to form a matrix of the appropriate type as the sum, difference etc. of two matrices which have the same multiplication.
IsLieMatrix(
mat ) C
A Lie matrix is a matrix whose multiplication is given by the
Lie bracket.
(Note that a matrix with ordinary matrix multiplication is in the
category IsOrdinaryMatrix
, see IsOrdinaryMatrix.)
Each matrix created by LieObject
is in the category IsLieMatrix
,
and arithmetic operations with objects in IsLieMatrix
produce
again matrices in IsLieMatrix
.
InfoMatrix V
The info class for matrix operations is InfoMatrix
.
mat +
scalar O
scalar +
mat O
returns 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 O
returns 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 O
scalar -
mat O
mat1 -
mat2
The definition for the -
operator are similar to the above definitions
for the +
operator, except that -
subtracts of course.
mat *
scalar O
scalar *
mat O
returns 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 O
returns the product of the vector vec and the matrix
mat. (This is the standard product of a row vector with a matrix.)
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 O
returns the product of the matrix mat and the vector
vec. (This is the standard product of a matrix with a column vector.)
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 O
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.
Inverse(
mat) O
returns the inverse matrix of mat, which must be an invertible square
matrix. If the matrix is not invertible, fail
is returned.
mat1 /
mat2 O
scalar /
mat O
mat /
scalar O
vec /
mat O
In general left
/
right is defined as
left
*
right^-1
.
Thus in the above forms the right operand must always be invertable.
mat ^
int O
returns 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 One(
mat)
, even if mat is not invertible.
mat1 ^
mat2 O
returns the conjugate of the matrix mat1 by the matrix
mat2, i.e. mat2
^-1 *
mat1 *
mat2. mat2 must be
invertible and mat1 must be a square matrix of the same dimension.
vec ^
mat O
This is in every respect equivalent to vec
*
mat. This
operations reflects the fact that matrices operate naturally on the
vector space by multiplication from the right.
scalar +
matlist O
matlist +
scalar O
scalar -
matlist O
matlist -
scalar O
scalar *
matlist O
matlist *
scalar O
matlist /
scalar O
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 O
matlist *
mat O
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 O
This form evaluates to matlist
*
mat^-1
. mat must of course
be invertable.
vec *
matlist O
returns the product of the vector vec and the list of
matrices mat. The lengths 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
ring. The product is the sum over vec
[
i] *
matlist[
i]
with
i running from 1 to l.
Comm(
mat1,
mat2 ) O
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
* One(
GF)
, where GF is the
finite field (see Operations for Finite Field Elements).
For all the above operations the result is a new mutable list, unless both operands are immutable, in which case the result is immutable.
DimensionsMat(
mat ) A
is a list of length 2, the first being the number of rows, the second being the number of columns of the matrix mat.
gap> DimensionsMat([[1,2,3],[4,5,6]]); [ 2, 3 ]
DefaultFieldOfMatrix(
mat ) A
is a field (not necessarily the smallest one) containing all entries of the matrix mat.
gap> DefaultFieldOfMatrix([[Z(4),Z(8)]]); GF(2^6)
TraceMat(
mat ) F
Trace(
mat ) F
The trace of a square matrix is the sum of its diagonal entries.
gap> TraceMat([[1,2,3],[4,5,6],[7,8,9]]); 15
DeterminantMat(
mat ) A
Determinant(
mat ) F
returns the determinant of the square matrix mat
gap> DeterminantMat([[1,2],[2,1]]); -3
IsMonomialMatrix(
mat ) P
A matrix is monomial if and only if it has exactly one nonzero entry in every row and every column.
gap> IsMonomialMatrix([[0,1],[1,0]]); true
IsDiagonalMat(
mat ) O
returns true if mat has only zero entries off the main diagonal, false otherwise.
IsUpperTriangularMat(
mat ) O
returns true if mat has only zero entries below the main diagonal, false otherwise.
IsLowerTriangularMat(
mat ) O
returns true if mat has only zero entries below the main diagonal, false otherwise.
IdentityMat(
m[,
F] ) F
returns a (mutable) m ×m identity matrix over the field given by F (i.e. the smallest field containing the element F or F itself if it is a field).
NullMat(
m,
n[,
F] ) F
returns a (mutable) m ×n null matrix over the field given by F.
gap> IdentityMat(3,1); [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] gap> NullMat(3,2,Z(3)); [ [ 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3) ] ]
EmptyMatrix(
char ) F
is an empty (ordinary) matrix in characteristic char that can be added
to or multiplied with empty lists (representing zero-dimensional row
vectors). It also acts (via ^
) on empty lists.
gap> EmptyMatrix(5); EmptyMatrix( 5 ) gap> AsList(last); [ ]
DiagonalMat(
vector ) F
returns a diagonal matrix mat with the diagonal entries given by vector.
gap> DiagonalMat([1,2,3]); [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 3 ] ]
PermutationMat(
perm,
dim[,
F] ) F
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.
gap> PermutationMat((1,2,3),4,1); [ [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 1 ] ]
TransposedMat(
mat ) A
TransposedMat
returns the transposed of the matrix mat, i.e., a new
matrix trans such that trans
[
i][
k] =
mat[
k][
i]
. As it is
an attribute it returns an immutable matrix.
MutableTransposedMat(
mat ) F
MutableTransposedMat
returns the transposed of the matrix mat as a
mutable matrix, i.e., a new matrix trans such that trans
[
i][
k] =
mat[
k][
i]
.
gap> TransposedMat([[1,2,3],[4,5,6],[7,8,9]]); [ [ 1, 4, 7 ], [ 2, 5, 8 ], [ 3, 6, 9 ] ]
KroneckerProduct(
mat1,
mat2 ) O
The Kronecker product of two matrices is the matrix obtained when
replacing each entry a of mat1 by the product a
*
mat2 in one
matrix.
gap> KroneckerProduct([[1,2]],[[5,7],[9,2]]); [ [ 5, 7, 10, 14 ], [ 9, 2, 18, 4 ] ]
ReflectionMat(
coeffs ) F
ReflectionMat(
coeffs,
root ) F
ReflectionMat(
coeffs,
conj ) F
ReflectionMat(
coeffs,
conj,
root ) F
Let coeffs be a row vector.
ReflectionMat
returns the matrix of the reflection in this vector.
More precisely, if coeffs is the coefficients of a vector v w.r.t. a basis B (see nowhere), say, then the returned matrix describes the reflection in v w.r.t. B as a map on a row space, with action from the right.
The optional argument root is a root of unity that determines the order of the reflection. The default is a reflection of order 2. For triflections one should choose a third root of unity etc. (see nowhere).
conj is a function of one argument that conjugates a ring element.
The default is ComplexConjugate
.
The matrix of the reflection in v is defined as
|
w = root
,
n is the length of the coefficient list,
and [`]
denotes the conjugation.
PrintArray(
array ) F
pretty-prints the array array.
MutableIdentityMat(
m[,
F] ) F
returns a (mutable) m ×m identity matrix over the field given
by F.
This is identical to IdentityMat
and is present in GAP 4.1
only for the sake of compatibility with beta-releases.
It should not be used in new code.
MutableNullMat(
m,
n[,
F] ) F
returns a (mutable) m ×n null matrix over the field given
by F.
This is identical to NullMat
and is present in GAP 4.1
only for the sake of compatibility with beta-releases.
It should not be used in new code.
RandomMat(
m,
n[,
R] ) F
RandomMat
returns a new mutable random matrix with m rows and
n columns with elements taken from the ring R, which defaults
to Integers
.
RandomInvertibleMat(
m[,
R] ) F
RandomInvertibleMat
returns a new mutable invertible random
matrix with m rows and columns with elements taken from the ring
R, which defaults to Integers
.
RandomUnimodularMat(
m ) F
returns a new random mutable m ×m matrix with integer entries that is invertible over the integers.
gap> RandomMat(2,3,GF(3)); [ [ Z(3)^0, Z(3), Z(3)^0 ], [ Z(3), Z(3)^0, Z(3)^0 ] ] gap> RandomInvertibleMat(4); [ [ -1, 0, 1, -1 ], [ 2, 1, 3, 0 ], [ 1, 4, 0, 2 ], [ -3, 2, 1, 0 ] ]
23.5 Matrices Representing Linear Equations and the Gaussian Algorithm
RankMat(
mat ) A
If mat is a matrix whose rows span a free module over the ring
generated by the matrix entries and their inverses
then RankMat
returns the dimension of this free module.
Otherwise fail
is returned.
Note that RankMat
may perform a Gaussian elimination.
For large rational matrices this may take very long,
because the entries may become very large.
gap> mat:=[[1,2,3],[4,5,6],[7,8,9]];; gap> RankMat(mat); 2
TriangulizeMat(
mat ) O
applies the Gaussian Algorithm to the mutable matrix mat and changes mat such that it is in upper triangular normal form (sometimes called ``Hermite normal form'').
gap> m:=MutableTransposedMat(mat); [ [ 1, 4, 7 ], [ 2, 5, 8 ], [ 3, 6, 9 ] ] gap> TriangulizeMat(m);m; [ [ 1, 0, -1 ], [ 0, 1, 2 ], [ 0, 0, 0 ] ]
NullspaceMat(
mat ) A
TriangulizedNullspaceMat(
mat ) A
returns a list of row vectors that form a basis of the vector space of
solutions to the equation vec
*
mat=0
. The result is an immutable
matrix. This basis is not guaranteed to be in any specific form.
The variant TriangulizedNullspaceMat
returns a basis of the nullspace
in triangulized form as is often needed for algorithms.
gap> mat:=[[1,2,3],[4,5,6],[7,8,9]];; gap> NullspaceMat(mat); [ [ 1, -2, 1 ] ]
SolutionMat(
mat,
vec ) O
returns a rwo vector x that is a solution of the equation x
*
mat
=
vec. It returns
fail
if no such vector exists.
gap> mat:=[[1,2,3],[4,5,6],[7,8,9]];; gap> SolutionMat(mat,[3,5,7]); [ 5/3, 1/3, 0 ]
BaseFixedSpace(
mats ) F
BaseFixedSpace
returns a list of row vectors that form a base of the
vector space V such that M v = v for all v in V and all matrices
M in the list mats. (This is the common eigenspace of all matrices
in mats for the eigenvalue 1.)
gap> BaseFixedSpace([[[1,2],[0,1]]]); [ [ 0, 1 ] ]
ElementaryDivisorsMat( [
ring, ]
mat ) O
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.
The operations are performed over the ring ring, which must contain
all matrix entries. For compatibility reasons it can be omitted and
defaults to Integers
.
gap> mat:=[[1,2,3],[4,5,6],[7,8,9]];; gap> ElementaryDivisorsMat(mat); [ 1, 3, 0 ]
DiagonalizeMat(
ring,
mat ) O
brings the mutable matrix mat, considered as a matrix over ring, into diagonal form by elementary row and column operations.
gap> m:=[[1,2],[2,1]];; gap> DiagonalizeMat(Integers,m);m; [ [ 1, 0 ], [ 0, 3 ] ]
See also chapter Integral Matrices and Lattices
SemiEchelonMat(
mat ) A
A matrix over a field F is in semi-echelon form if the first nonzero element in each row is the identity of F, and all values exactly below these pivots are the zero of F.
SemiEchelonMat
returns a record that contains information about
a semi-echelonized form of the matrix mat.
The components of this record are
vectors
heads
SemiEchelonMatTransformation(
mat ) A
does the same as SemiEchelonMat
but additionally stores the linear
transformation T performed on the matrix.
The additional components of the result are
coeffs
vectors
component,
with respect to the rows of mat, that is, coeffs * mat
is the vectors
component.
relations
gap> SemiEchelonMatTransformation([[1,2,3],[0,0,1]]); rec( heads := [ 1, 0, 2 ], vectors := [ [ 1, 2, 3 ], [ 0, 0, 1 ] ], coeffs := [ [ 1, 0 ], [ 0, 1 ] ], relations := [ ] )
SemiEchelonMats(
mats ) A
A list of matrices over a field F is in semi-echelon form if the list of row vectors obtained on concatenating the rows of each matrix is a semi-echelonized matrix (see SemiEchelonMat).
SemiEchelonMats
returns a record that contains information about
a semi-echelonized form of the list mats of matrices.
The components of this record are
vectors
heads
BaseMat(
mat ) A
returns a basis for the row space generated by the rows of mat in the form of an immutable matrix.
gap> BaseMat(mat); [ [ 1, 0, -1 ], [ 0, 1, 2 ] ]
BaseOrthogonalSpaceMat(
mat ) A
Let V be the row space generated by the rows of mat (over any field
that contains all entries of mat). BaseOrthogonalSpaceMat(
mat )
computes a base of the orthogonal space of V.
The rows of mat need not be linearly independent.
SumIntersectionMat(
M1,
M2 ) O
performs Zassenhaus' algorithm to compute bases for the sum and the intersection of spaces generated by the rows of the matrices M1, M2.
returns a list of length 2, at first position a base of the sum, at second position a base of the intersection. Both bases are in semi-echelon form (see Echelonized matrices).
gap> SumIntersectionMat(mat,[[2,7,6],[5,9,4]]); [ [ [ 1, 2, 3 ], [ 0, 1, 2 ], [ 0, 0, 1 ] ], [ [ 1, -3/4, -5/2 ] ] ]
BaseSteinitzVectors(
bas,
mat ) F
find vectors extending mat to a basis spanning the span of bas. Both bas and mat must be matrices of full (row) rank. It returns a record with the following components:
subspace
factorspace
factorzero
heads
subspace
and factorspace
.
gap> BaseSteinitzVectors(IdentityMat(3,1),[[11,13,15]]); rec( factorspace := [ [ 0, 1, 15/13 ], [ 0, 0, 1 ] ], factorzero := [ 0, 0, 0 ], subspace := [ [ 1, 13/11, 15/11 ] ], heads := [ -1, 1, 2 ] )
See also chapter Integral Matrices and Lattices
DiagonalOfMat(
mat ) O
returns the diagonal of mat as a list.
gap> DiagonalOfMat([[1,2],[3,4]]); [ 1, 4 ]
UpperSubdiagonal(
mat,
pos ) O
returns a mutable list containing the entries of the posth upper subdiagonal of mat.
gap> UpperSubdiagonal(mat,1); [ 2, 6 ]
DepthOfUpperTriangularMatrix(
mat ) A
If mat is an upper triangular matrix this attribute returns the index of the first nonzero diagonal.
gap> DepthOfUpperTriangularMatrix([[0,1,2],[0,0,1],[0,0,0]]); 1 gap> DepthOfUpperTriangularMatrix([[0,0,2],[0,0,0],[0,0,0]]); 2
JordanDecomposition(
mat ) A
JordanDecomposition(
mat )
returns a list [S,N]
such that
S
is a semisimple matrix and N
is nilpotent. Furthermore, S
and N
commute and mat
=S+N
.
gap> mat:=[[1,2,3],[4,5,6],[7,8,9]];; gap> JordanDecomposition(mat); [ [ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ], [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ]
BlownUpMat(
B,
mat ) F
Let B be a basis of a field extension F / K,
and mat a matrix whose entries are all in F.
(This is not checked.)
BlownUpMat
returns a matrix over K that is obtained by replacing each
entry of mat by its regular representation w.r.t. B.
More precisely, regard mat as the matrix of a linear transformation on the row space Fn w.r.t. the F-basis with vectors (v1, ldots, vn), say, and suppose that the basis B consists of the vectors (b1, ¼, bm); then the returned matrix is the matrix of the linear transformation on the row space Kmn w.r.t. the K-basis whose vectors are (b1 v1, ¼bm v1, ¼, bm vn).
Note that the linear transformations act on row vectors, i.e., each row of the matrix is a concatenation of vectors of B-coefficients.
BlownUpVector(
B,
vector ) F
Let B be a basis of a field extension F / K,
and vector a row vector whose entries are all in F.
BlownUpVector
returns a row vector over K that is obtained by
replacing each entry of vector by its coefficients w.r.t. B.
So BlownUpVector
and BlownUpMat
(see BlownUpMat) are compatible
in the sense that for a matrix mat over F,
BlownUpVector(
B,
mat *
vector )
is equal to
BlownUpMat(
B,
mat ) * BlownUpVector(
B,
vector )
.
gap> B:= Basis( CF(4), [ 1, E(4) ] );; gap> mat:= [ [ 1, E(4) ], [ 0, 1 ] ];; vec:= [ 1, E(4) ];; gap> bmat:= BlownUpMat( B, mat );; bvec:= BlownUpVector( B, vec );; gap> Display( bmat ); bvec; [ [ 1, 0, 0, 1 ], [ 0, 1, -1, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ] [ 1, 0, 0, 1 ] gap> bvec * bmat = BlownUpVector( B, vec * mat ); true
CompanionMat(
poly ) F
computes a companion matrix of the polynomial poly. This matrix has poly as its minimal polynomial.
23.11 Matrices over Finite Fields
Just as for row vectors, (see section Row Vectors over Finite Fields), GAP has a special representation for matrices over small finite fields.
To be eligible to be represented in this way, each row of a matrix must be immutable, and able to be represented as a compact row vector over the same finite field.
gap> v := Z(2)*[1,0,0,1,1]; [ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ] gap> ConvertToVectorRep(v,2); 2 gap> v; <a GF2 vector of length 5> gap> MakeImmutable(v); gap> m := [v]; <a 1x5 matrix over GF2> gap> m := [v,v]; <a 2x5 matrix over GF2> gap> m := [v,v,v]; <a 3x5 matrix over GF2> gap> v := Z(3)*[1..8]; [ Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3)^0 ] gap> ConvertToVectorRep(v); 3 gap> MakeImmutable(v); gap> m := [v]; [ [ Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3)^0 ] ] gap> RepresentationsOfObject(m); [ "IsPositionalObjectRep", "Is8BitMatrixRep" ] gap> m := [v,v,v,v]; < mutable compressed matrix 4x8 over GF(3) >
All compressed matrices over GF(2) are viewed as <a
nx
m matrix
over GF2>
, while over fields GF(q) for q between 3 and 256, matrices
with 25 or more entries are views in this way, and smaller ones as
lists of lists.
Under some circumstances, as we saw above, GAP can detect that a matrix is eligible to be represented in this way, and automatically does so. For other cases there is a function:
ConvertToMatrixRep(
list ) F
ConvertToMatrixRep(
list,
field ) F
ConvertToMatrixRep(
list,
fieldsize ) F
ConvertToMatrixRep(
list )
converts list to an internal
matrix representation if possible. ConvertToMatrixRep(
list ,
field )
converts list to an internal matrix representation
appropriate for a matrix over field. It is forbidden to call
this function unless all elements of list are vectors with
entries in field.
Instead of a field also its size fieldsize may be given.
list may already be a compressed matrix. In this case, if no field or fieldsize is given, then nothing happens.
list itself may be mutable, but its entries must be immutable.
The return value is the size of the field over which the matrix
ends up written, if it is written in a compressed representation.
Otherwise it is fail
.
Note that the main advantage of this special representation of matrices is in low dimensions, where various overheads can be reduced. In higher dimensions, a list of compressed vectors will be almost as fast. Note also that list access and assignment will be somewhat slower for compressed matrices than for plain lists.
Finally note that a vector cannot be shared between two compressed
matrices over different fields. To avoid risk of this, when a vector
becomes part of a compressed matrix, the filter
IsLockedRepresentationVector
is set for it, and neither
ConvertToVectorRep
nor ConvertToMatrixRep
will ever alter a vector
with this filter.
gap> v := [Z(2),Z(2)]; <a GF2 vector of length 2> gap> m := [v,v]; # v is mutable, so m cannot be compressed [ <a GF2 vector of length 2>, <a GF2 vector of length 2> ] gap> ConvertToMatrixRep(m); # even if we ask for it fail gap> m; [ <a GF2 vector of length 2>, <a GF2 vector of length 2> ] gap> mi := List(m, Immutable); # but if we make the rows immutable <a 2x2 matrix over GF2> gap> m2 := [mi[1], [Z(4),Z(4)]]; # now try and mix in some GF(4) [ <an immutable GF2 vector of length 2>, [ Z(2^2), Z(2^2) ] ] gap> ConvertToMatrixRep(m2); # bur m2[1] is locked fail gap> m2 := [ShallowCopy(mi[1]), [Z(4),Z(4)]]; # a fresh copy f row 1 [ <a GF2 vector of length 2>, [ Z(2^2), Z(2^2) ] ] gap> MakeImmutable(m2[1]); gap> MakeImmutable(m2[2]); gap> ConvertToMatrixRep(m2); # now it works 4 gap> m2; [ [ Z(2)^0, Z(2)^0 ], [ Z(2^2), Z(2^2) ] ] gap> RepresentationsOfObject(m2); [ "IsPositionalObjectRep", "Is8BitMatrixRep" ] gap>
There are also two operations that are only available for matrices written over finite fields.
ProjectiveOrder(
mat ) A
Returns an integer n and a finite field element e such that A^n = eI. mat must be a matrix defined over a finite field.
gap> ProjectiveOrder([[1,4],[5,2]]*Z(11)^0); [ 5, Z(11)^5 ]
SimultaneousEigenvalues(
matlist,
expo ) F
The matrices in matlist must be matrices over GF(q) for some
prime q. Together, they must generate an abelian p-group of
exponent expo.
Then the eigenvalues of mat in the splitting field GF(
q^
r)
for
some r are powers of an element x in the splitting field, which is
of order expo. SimultaneousEigenvalues
returns a matrix of
integers mod expo, say (ai,j), such that the power
xai,j is an eigenvalue of the i-th matrix in matlist and
the eigenspaces of the different matrices to the eigenvalues
xai,j for fixed j are equal.
Finally, there are two operations that deal with matrices over a ring, but only care about the residues of their entries modulo some ring element. In the case of the integers and a prime number, this is effectively computation in a matrix over a finite field.
InverseMatMod(
mat,
obj ) O
For a square matrix mat, InverseMatMod
returns a matrix inv
such that inv
*
mat is congruent to the identity matrix modulo
obj, if such a matrix exists, and
fail
otherwise.
gap> mat:= [ [ 1, 2 ], [ 3, 4 ] ];; inv:= InverseMatMod( mat, 5 ); [ [ 3, 1 ], [ 4, 2 ] ] gap> mat * inv; [ [ 11, 5 ], [ 25, 11 ] ]
NullspaceModQ(
E,
q ) F
E must be a matrix of integers and q a prime power.
Then NullspaceModQ
returns the set of all vectors of integers modulo
q, which solve the homogeneous equation system given by E modulo q.
gap> mat:= [ [ 1, 3 ], [ 1, 2 ], [ 1, 1 ] ];; NullspaceModQ( mat, 5 ); [ [ 0, 0, 0 ], [ 1, 3, 1 ], [ 2, 1, 2 ], [ 4, 2, 4 ], [ 3, 4, 3 ] ]
Block matrices are a special representation of matrices which can save a
lot of memory if large matrices have a block structure with lots of zero
blocks. GAP uses the representation IsBlockMatrixRep
to store block
matrices.
AsBlockMatrix(
m,
nrb,
ncb ) F
returns a block matrix with nrb row blocks and ncb column blocks which is equal to the ordinary matrix m.
BlockMatrix(
blocks,
nrb,
ncb ) F
BlockMatrix(
blocks,
nrb,
ncb,
rpb,
cpb,
zero ) F
BlockMatrix
returns an immutable matrix in the sparse representation
IsBlockMatrixRep
.
The nonzero blocks are described by the list blocks of triples,
the matrix has nrb row blocks and ncb column blocks.
If blocks is empty (i.e., if the matrix is a zero matrix) then the dimensions of the blocks must be entered as rpb and cpb, and the zero element as zero.
Note that all blocks must be ordinary matrices (see IsOrdinaryMatrix), and also the block matrix is an ordinary matrix.
MatrixByBlockMatrix(
blockmat ) F
returns an ordinary matrix that is equal to the block matrix blockmat.
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GAP 4 manual