24.4 Matrix Constructions

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).

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) ] ]

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);
[  ]

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 ] ]

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 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 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].

If mat is a mutable matrix, then the transposed is computed by swapping the entries in mat. In this way mat gets changed. In all other cases the transposed is computed by TransposedMat.

gap> TransposedMat([[1,2,3],[4,5,6],[7,8,9]]);
[ [ 1, 4, 7 ], [ 2, 5, 8 ], [ 3, 6, 9 ] ]
gap> mm:= [[1,2,3],[4,5,6],[7,8,9]];;
gap> TransposedMatDestructive( mm );
[ [ 1, 4, 7 ], [ 2, 5, 8 ], [ 3, 6, 9 ] ]
gap> mm;
[ [ 1, 4, 7 ], [ 2, 5, 8 ], [ 3, 6, 9 ] ]

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 ] ]

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

pretty-prints the array array.

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.

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.

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