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24 Matrices

In **GAP**, Matrices can be represented by lists of row vectors, see 23. (For a more general way to represent vectors and matrices, see Chapter 26). The row vectors must all have the same length, and their elements must lie in a common ring. However, since checking rectangularness can be expensive functions and methods of operations for matrices often will not give an error message for non-rectangular lists of lists –in such cases the result is undefined.

Because matrices are just a special case of lists, all operations and functions for lists are applicable to matrices also (see chapter 21). This especially includes accessing elements of a matrix (see 21.3), changing elements of a matrix (see 21.4), and comparing matrices (see 21.10).

Note that, since a matrix is a list of lists, the behaviour of `ShallowCopy`

(12.7-1) for matrices is just a special case of `ShallowCopy`

(12.7-1) for lists (see 21.7); called with an immutable matrix `mat`, `ShallowCopy`

(12.7-1) returns a mutable matrix whose rows are identical to the rows of `mat`. In particular the rows are still immutable. To get a matrix whose rows are mutable, one can use `List( `

.`mat`, ShallowCopy )

`‣ InfoMatrix` | ( info class ) |

The info class for matrix operations is `InfoMatrix`

.

`‣ IsMatrix` ( obj ) | ( category ) |

By convention *matrix* is a list of lists of equal length whose entries lie in a common ring.

For technical reasons laid out at the top of Chapter 24, the filter `IsMatrix`

is a synonym for a table of ring elements, (see `IsTable`

(21.1-4) and `IsRingElement`

(31.14-16)). This means that `IsMatrix`

returns `true`

for tables such as `[[1,2],[3]]`

. If necessary, `IsRectangularTable`

(21.1-5) can be used to test whether an object is a list of homogenous lists of equal lengths manually.

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`

(24.2-2) and `IsLieMatrix`

(24.2-3) 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 gap> mat:=[[1,2],[3]]; [ [ 1, 2 ], [ 3 ] ] gap> IsMatrix(mat); true gap> IsRectangularTable(mat); false

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`

(24.5-3) below.

gap> [[0]]*[[]]; [ [ ] ] gap> IsMatrix([[]]); false

`‣ IsOrdinaryMatrix` ( obj ) | ( category ) |

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 ) | ( category ) |

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`

(24.2-2).)

Each matrix created by `LieObject`

(64.1-1) is in the category `IsLieMatrix`

, and arithmetic operations with objects in `IsLieMatrix`

produce again matrices in `IsLieMatrix`

.

The rules for arithmetic operations involving matrices are in fact special cases of those for the arithmetic of lists, given in Section 21.11 and the following sections, here we reiterate that definition, in the language of vectors and matrices.

Note that the additive behaviour sketched below is defined only for lists in the category `IsGeneralizedRowVector`

(21.12-1), and the multiplicative behaviour is defined only for lists in the category `IsMultiplicativeGeneralizedRowVector`

(21.12-2) (see 21.12).

`mat1` + `mat2`

returns the sum of the two matrices `mat1` and `mat2`, Probably the most usual situation is that `mat1` and `mat2` have the same dimensions and are defined over a common field; in this case the sum is a new matrix over the same field where each entry is the sum of the corresponding entries of the matrices.

In more general situations, the sum of two matrices need not be a matrix, for example adding an integer matrix `mat1` and a matrix `mat2` over a finite field yields the table of pointwise sums, which will be a mixture of finite field elements and integers if `mat1` has bigger dimensions than `mat2`.

`scalar` + `mat`

`mat` + `scalar`

returns the sum of the scalar `scalar` and the matrix `mat`. Probably the most usual situation is that the entries of `mat` lie in a common field with `scalar`; in this case the sum is a new matrix over the same field where each entry is the sum of the scalar and the corresponding entry of the matrix.

More general situations are for example the sum of an integer scalar and a matrix over a finite field, or the sum of a finite field element and an integer matrix.

`mat1` - `mat2`

`scalar` - `mat`

`mat` - `scalar`

Subtracting a matrix or scalar is defined as adding its additive inverse, so the statements for the addition hold likewise.

`scalar` * `mat`

`mat` * `scalar`

returns the product of the scalar `scalar` and the matrix `mat`. Probably the most usual situation is that the elements of `mat` lie in a common field with `scalar`; in this case the product is a new matrix over the same field where each entry is the product of the scalar and the corresponding entry of the matrix.

More general situations are for example the product of an integer scalar and a matrix over a finite field, or the product of a finite field element and an integer matrix.

`vec` * `mat`

returns the product of the row vector `vec` and the matrix `mat`. Probably the most usual situation is that `vec` and `mat` have the same lengths and are defined over a common field, and that all rows of `mat` have some common length \(m\); in this case the product is a new row vector of length \(m\) over the same field which is the sum of the scalar multiples of the rows of `mat` with the corresponding entries of `vec`.

More general situations are for example the product of an integer vector and a matrix over a finite field, or the product of a vector over a finite field and an integer matrix.

`mat` * `vec`

returns the product of the matrix `mat` and the row vector `vec`. (This is the standard product of a matrix with a *column* vector.) Probably the most usual situation is that the length of `vec` and of all rows of `mat` are equal, and that the elements of `mat` and `vec` lie in a common field; in this case the product is a new row vector of the same length as `mat` and over the same field which is the sum of the scalar multiples of the columns of `mat` with the corresponding entries of `vec`.

More general situations are for example the product of an integer matrix and a vector over a finite field, or the product of a matrix over a finite field and an integer vector.

`mat1` * `mat2`

This form evaluates to the (Cauchy) product of the two matrices `mat1` and `mat2`. Probably the most usual situation is that the number of columns of `mat1` equals the number of rows of `mat2`, and that the elements of `mat` and `vec` 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 \(j\) of the product is the sum of \(\textit{mat1}[i][l] * \textit{mat2}[l][j]\), with \(l\) running from \(1\) to \(n\).

`Inverse( `

`mat` )

returns the inverse of the matrix `mat`, which must be an invertible square matrix. If `mat` is not invertible then `fail`

is returned.

`mat1` / `mat2`

`scalar` / `mat`

`mat` / `scalar`

`vec` / `mat`

In general,

is defined as `left` / `right`

. Thus in the above forms the right operand must always be invertible.`left` * `right`^-1

`mat` ^ `int`

`mat1` ^ `mat2`

`vec` ^ `mat`

Powering a square matrix `mat` by an integer `int` yields the `int`-th power of `mat`; if `int` is negative then `mat` must be invertible, if `int` is `0`

then the result is the identity matrix `One( `

, even if `mat` )`mat` is not invertible.

Powering a square matrix `mat1` by an invertible square matrix `mat2` of the same dimensions yields the conjugate of `mat1` by `mat2`, i.e., the matrix

.`mat2`^-1 * `mat1` * `mat2`

Powering a row vector `vec` by a matrix `mat` is in every respect equivalent to

. This operations reflects the fact that matrices act naturally on row vectors by multiplication from the right, and that the powering operator is `vec` * `mat`**GAP**'s standard for group actions.

`Comm( `

`mat1`, `mat2` )

returns the commutator of the square invertible matrices `mat1` and `mat2` of the same dimensions and over a common field, which is the matrix

.`mat1`^-1 * `mat2`^-1 * `mat1` * `mat2`

The following cases are still special cases of the general list arithmetic defined in 21.11.

`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 divided into a list `matlist` of matrices. 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 list `matlist` of matrices. The result is a new list of matrices, where each entry is the product of `mat` and the corresponding entry in `matlist`.

`matlist` / `mat`

Dividing a list `matlist` of matrices by an invertible matrix `mat` evaluates to

.`matlist` * `mat`^-1

`vec` * `matlist`

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

with `vec`[`i`] * `matlist`[`i`]`i` running from 1 to `l`.

For the mutability of results of arithmetic operations, see 12.6.

`‣ DimensionsMat` ( mat ) | ( attribute ) |

is a list of length 2, the first being the number of rows, the second being the number of columns of the matrix `mat`. If `mat` is malformed, that is, it is not a `IsRectangularTable`

(21.1-5), returns `fail`

.

gap> DimensionsMat([[1,2,3],[4,5,6]]); [ 2, 3 ] gap> DimensionsMat([[1,2,3],[4,5]]); fail

`‣ DefaultFieldOfMatrix` ( mat ) | ( attribute ) |

For a matrix `mat`, `DefaultFieldOfMatrix`

returns either a field (not necessarily the smallest one) containing all entries of `mat`, or `fail`

.

If `mat` is a matrix of finite field elements or a matrix of cyclotomics, `DefaultFieldOfMatrix`

returns the default field generated by the matrix entries (see 59.3 and 18.1).

gap> DefaultFieldOfMatrix([[Z(4),Z(8)]]); GF(2^6)

`‣ TraceMatrix` ( mat ) | ( attribute ) |

`‣ TraceMat` ( mat ) | ( attribute ) |

`‣ Trace` ( mat ) | ( attribute ) |

The trace of a square matrix is the sum of its diagonal entries.

gap> TraceMatrix([[1,2,3],[4,5,6],[7,8,9]]); 15

`‣ DeterminantMatrix` ( mat ) | ( attribute ) |

`‣ DeterminantMat` ( mat ) | ( attribute ) |

`‣ Determinant` ( mat ) | ( operation ) |

returns the determinant of the square matrix `mat`.

These methods assume implicitly that `mat` is defined over an integral domain whose quotient field is implemented in **GAP**. For matrices defined over an arbitrary commutative ring with one see `DeterminantMatDivFree`

(24.4-6).

`‣ DeterminantMatrixDestructive` ( mat ) | ( operation ) |

`‣ DeterminantMatDestructive` ( mat ) | ( operation ) |

Does the same as `DeterminantMatrix`

(24.4-4), with the difference that it may destroy its argument. The matrix `mat` must be mutable.

gap> DeterminantMatrix([[1,2],[2,1]]); -3 gap> mm:= [[1,2],[2,1]];; gap> DeterminantMatrixDestructive( mm ); -3 gap> mm; [ [ 1, 2 ], [ 0, -3 ] ]

`‣ DeterminantMatrixDivFree` ( mat ) | ( operation ) |

`‣ DeterminantMatDivFree` ( mat ) | ( operation ) |

return the determinant of a square matrix `mat` over an arbitrary commutative ring with one using the division free method of Mahajan and Vinay [MV97].

`‣ IsEmptyMatrix` ( M ) | ( property ) |

Returns: A boolean

Is `true`

if the matrix object `M` either has zero columns or zero rows, and `false`

otherwise. In other words, a matrix object is empty if it has no entries.

`‣ IsMonomialMatrix` ( mat ) | ( property ) |

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

`‣ IsDiagonalMatrix` ( mat ) | ( property ) |

`‣ IsDiagonalMat` ( mat ) | ( property ) |

return `true`

if the matrix `mat` has only zero entries off the main diagonal, and `false`

otherwise.

gap> IsDiagonalMatrix( [ [ 1 ] ] ); true gap> IsDiagonalMatrix( [ [ 1, 0, 0 ], [ 0, 1, 0 ] ] ); true gap> IsDiagonalMatrix( [ [ 0, 1 ], [ 1, 0 ] ] ); false

`‣ IsUpperTriangularMatrix` ( mat ) | ( property ) |

`‣ IsUpperTriangularMat` ( mat ) | ( property ) |

return `true`

if the matrix `mat` has only zero entries below the main diagonal, and `false`

otherwise.

gap> IsUpperTriangularMatrix( [ [ 1 ] ] ); true gap> IsUpperTriangularMatrix( [ [ 1, 2, 3 ], [ 0, 5, 6 ] ] ); true gap> IsUpperTriangularMatrix( [ [ 0, 1 ], [ 1, 0 ] ] ); false

`‣ IsLowerTriangularMatrix` ( mat ) | ( property ) |

`‣ IsLowerTriangularMat` ( mat ) | ( property ) |

return `true`

if the matrix `mat` has only zero entries above the main diagonal, and `false`

otherwise.

gap> IsLowerTriangularMatrix( [ [ 1 ] ] ); true gap> IsLowerTriangularMatrix( [ [ 1, 0, 0 ], [ 2, 3, 0 ] ] ); true gap> IsLowerTriangularMatrix( [ [ 0, 1 ], [ 1, 0 ] ] ); false

`‣ IdentityMat` ( m[, R] ) | ( function ) |

returns a (mutable) `m`\(\times\)`m` identity matrix over the ring given by `R`. Here, `R` can be either a ring, or an element of a ring. By default, an integer matrix is created.

gap> IdentityMat(3); [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] gap> IdentityMat(2,Integers mod 15); [ [ ZmodnZObj( 1, 15 ), ZmodnZObj( 0, 15 ) ], [ ZmodnZObj( 0, 15 ), ZmodnZObj( 1, 15 ) ] ] gap> IdentityMat(2,Z(3)); [ [ Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3)^0 ] ]

`‣ NullMat` ( m, n[, R] ) | ( function ) |

returns a (mutable) `m`\(\times\)`n` null matrix over the ring given by by `R`. Here, `R` can be either a ring, or an element of a ring. By default, an integer matrix is created.

gap> NullMat(3,2); [ [ 0, 0 ], [ 0, 0 ], [ 0, 0 ] ] gap> NullMat(2,2,Integers mod 15); [ [ ZmodnZObj( 0, 15 ), ZmodnZObj( 0, 15 ) ], [ ZmodnZObj( 0, 15 ), ZmodnZObj( 0, 15 ) ] ] 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 ) | ( function ) |

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 the operation `\^`

(31.12-1)) on empty lists.

gap> EmptyMatrix(5); EmptyMatrix( 5 ) gap> AsList(last); [ ]

`‣ DiagonalMat` ( vector ) | ( function ) |

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] ) | ( 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.

gap> PermutationMat((1,2,3),4,1); [ [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 1 ] ]

`‣ TransposedMatImmutable` ( mat ) | ( attribute ) |

`‣ TransposedMatAttr` ( mat ) | ( attribute ) |

`‣ TransposedMat` ( mat ) | ( attribute ) |

`‣ TransposedMatMutable` ( mat ) | ( operation ) |

`‣ TransposedMatOp` ( mat ) | ( operation ) |

These functions all return the transposed of the matrix object `mat`, i.e., a matrix object \(trans\) such that \(trans[i,k] = \textit{mat}[k,i]\) holds.

They differ only w.r.t. the mutability of the result.

`TransposedMat`

is an attribute and hence returns an immutable result. `TransposedMatMutable`

is guaranteed to return a new *mutable* matrix.

`TransposedMatImmutable`

and `TransposedMatAttr`

are synonyms of `TransposedMat`

, and `TransposedMatOp`

is a synonym of `TransposedMatMutable`

, in analogy to operations such as `Zero`

(31.10-3).

`‣ TransposedMatDestructive` ( mat ) | ( operation ) |

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`

(24.5-6).

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 ] ]

`‣ KroneckerProduct` ( mat1, mat2 ) | ( operation ) |

The Kronecker product of two matrices is the matrix obtained when replacing each entry `a` of `mat1` by the product

in one matrix.`a`*`mat2`

gap> KroneckerProduct([[1,2]],[[5,7],[9,2]]); [ [ 5, 7, 10, 14 ], [ 9, 2, 18, 4 ] ]

`‣ ReflectionMat` ( coeffs[, conj][, root] ) | ( function ) |

Let `coeffs` be a row vector. `ReflectionMat`

returns the matrix of the reflection in this vector.

More precisely, if `coeffs` is the coefficients list of a vector \(v\) w.r.t. a basis \(B\) (see `Basis`

(61.5-2)) 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 `E`

(18.1-1)).

`conj` is a function of one argument that conjugates a ring element. The default is `ComplexConjugate`

(18.5-2).

The matrix of the reflection in \(v\) is defined as

\[ M = I_n + \overline{{v^{tr}}} \cdot (w-1) / ( v \overline{{v^{tr}}} ) \cdot v \]

where \(w\) equals `root`, \(n\) is the length of the coefficient list, and \(\overline{{\vphantom{x}}}\) denotes the conjugation.

So \(v\) is mapped to \(w v\), with default \(-v\), and any vector \(x\) with the property \(x \overline{{v^{tr}}} = 0\) is fixed by the reflection.

`‣ PrintArray` ( array ) | ( function ) |

pretty-prints the array `array`.

`‣ RandomMat` ( [rs, ]m, n[, R] ) | ( function ) |

`RandomMat`

returns a new mutable random matrix with `m` rows and `n` columns with elements taken from the ring `R`, which defaults to `Integers`

(14). Optionally, a random source `rs` can be supplied.

gap> RandomMat(2,3,GF(3)); [ [ Z(3), Z(3), 0*Z(3) ], [ Z(3), Z(3)^0, Z(3) ] ]

`‣ RandomInvertibleMat` ( [rs, ]m[, R] ) | ( function ) |

`RandomInvertibleMat`

returns a new mutable invertible random matrix with `m` rows and columns with elements taken from the ring `R`, which defaults to `Integers`

(14). Optionally, a random source `rs` can be supplied.

gap> m := RandomInvertibleMat(4); [ [ -4, 1, 0, -1 ], [ -1, -1, 1, -1 ], [ 1, -2, -1, -2 ], [ 0, -1, 2, -2 ] ] gap> m^-1; [ [ -1/8, -11/24, 1/24, 1/4 ], [ 1/4, -13/12, -1/12, 1/2 ], [ -1/8, 5/24, -7/24, 1/4 ], [ -1/4, 3/4, -1/4, -1/2 ] ]

`‣ RandomUnimodularMat` ( [rs, ]m ) | ( function ) |

returns a new random mutable `m`\(\times\)`m` matrix with integer entries that is invertible over the integers. Optionally, a random source `rs` can be supplied. If the option `domain` is given, random selection is made from `domain`, otherwise from `Integers`

gap> m := RandomUnimodularMat(3); [ [ -5, 1, 0 ], [ 12, -2, -1 ], [ -14, 3, 0 ] ] gap> m^-1; [ [ -3, 0, 1 ], [ -14, 0, 5 ], [ -8, -1, 2 ] ] gap> RandomUnimodularMat(3:domain:=[-1000..1000]); [ [ 312330173, 15560030349, -125721926670 ], [ -307290, -15309014, 123693281 ], [ -684293792, -34090949551, 275448039848 ] ]

`‣ RankMatrix` ( mat ) | ( attribute ) |

`‣ RankMat` ( mat ) | ( attribute ) |

If `mat` is a matrix object representing a matrix whose rows span a free module over the ring generated by the matrix entries and their inverses then `RankMatrix`

returns the dimension of this free module. Otherwise `fail`

is returned.

Note that `RankMatrix`

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> RankMatrix( mat ); 2

`‣ TriangulizedMat` ( mat ) | ( operation ) |

`‣ RREF` ( mat ) | ( operation ) |

Computes an upper triangular form of the matrix `mat` via the Gaussian Algorithm. It returns a mutable matrix in upper triangular form. This is sometimes also called "Hermite normal form" or "Reduced Row Echelon Form". `RREF`

is a synonym for `TriangulizedMat`

.

`‣ TriangulizeMat` ( mat ) | ( operation ) |

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" or "Reduced Row Echelon Form").

gap> m:=TransposedMatMutable(mat); [ [ 1, 4, 7 ], [ 2, 5, 8 ], [ 3, 6, 9 ] ] gap> TriangulizeMat(m);m; [ [ 1, 0, -1 ], [ 0, 1, 2 ], [ 0, 0, 0 ] ] gap> m:=TransposedMatMutable(mat); [ [ 1, 4, 7 ], [ 2, 5, 8 ], [ 3, 6, 9 ] ] gap> TriangulizedMat(m);m; [ [ 1, 0, -1 ], [ 0, 1, 2 ], [ 0, 0, 0 ] ] [ [ 1, 4, 7 ], [ 2, 5, 8 ], [ 3, 6, 9 ] ]

`‣ NullspaceMat` ( mat ) | ( attribute ) |

`‣ TriangulizedNullspaceMat` ( mat ) | ( attribute ) |

returns a list of row vectors that form a basis of the vector space of solutions to the equation

. The result is an immutable matrix. This basis is not guaranteed to be in any specific form.`vec`*`mat`=0

The variant `TriangulizedNullspaceMat`

returns a basis of the nullspace in triangulized form as is often needed for algorithms.

`‣ NullspaceMatDestructive` ( mat ) | ( operation ) |

`‣ TriangulizedNullspaceMatDestructive` ( mat ) | ( operation ) |

This function does the same as `NullspaceMat`

(24.7-4). However, the latter function makes a copy of `mat` to avoid having to change it. This function does not do that; it returns the nullspace and may destroy `mat`; this saves a lot of memory in case `mat` is big. The matrix `mat` must be mutable.

The variant `TriangulizedNullspaceMatDestructive`

returns a basis of the nullspace in triangulized form. It may destroy the matrix `mat`.

gap> mat:=[[1,2,3],[4,5,6],[7,8,9]];; gap> NullspaceMat(mat); [ [ 1, -2, 1 ] ] gap> mm:=[[1,2,3],[4,5,6],[7,8,9]];; gap> NullspaceMatDestructive( mm ); [ [ 1, -2, 1 ] ] gap> mm; [ [ 1, 2, 3 ], [ 0, -3, -6 ], [ 0, 0, 0 ] ]

`‣ SolutionMat` ( mat, vec ) | ( operation ) |

returns a row vector `x` that is a solution of the equation

. It returns `x` * `mat` = `vec``fail`

if no such vector exists.

`‣ SolutionMatDestructive` ( mat, vec ) | ( operation ) |

Does the same as `SolutionMat( `

except that it may destroy the matrix `mat`, `vec` )`mat` and the vector `vec`. The matrix `mat` must be mutable.

gap> mat:=[[1,2,3],[4,5,6],[7,8,9]];; gap> SolutionMat(mat,[3,5,7]); [ 5/3, 1/3, 0 ] gap> mm:= [[1,2,3],[4,5,6],[7,8,9]];; gap> v:= [3,5,7];; gap> SolutionMatDestructive( mm, v ); [ 5/3, 1/3, 0 ] gap> mm; [ [ 1, 2, 3 ], [ 0, -3, -6 ], [ 0, 0, 0 ] ] gap> v; [ 0, 0, 0 ]

`‣ BaseFixedSpace` ( mats ) | ( function ) |

`BaseFixedSpace`

returns a list of row vectors that form a base of the vector space \(V\) such that \(v M = v\) 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 ] ]

`‣ GeneralisedEigenvalues` ( F, A ) | ( operation ) |

`‣ GeneralizedEigenvalues` ( F, A ) | ( operation ) |

The generalised eigenvalues of the matrix `A` over the field `F`.

`‣ GeneralisedEigenspaces` ( F, A ) | ( operation ) |

`‣ GeneralizedEigenspaces` ( F, A ) | ( operation ) |

The generalised eigenspaces of the matrix `A` over the field `F`.

`‣ Eigenvalues` ( F, A ) | ( operation ) |

The eigenvalues of the matrix `A` over the field `F`.

`‣ Eigenspaces` ( F, A ) | ( operation ) |

The eigenspaces of the matrix `A` over the field `F`.

`‣ Eigenvectors` ( F, A ) | ( operation ) |

The eigenvectors of the matrix `A` over the field `F`.

See also chapter 25.

`‣ ElementaryDivisorsMat` ( [ring, ]mat ) | ( operation ) |

`‣ ElementaryDivisorsMatDestructive` ( ring, mat ) | ( function ) |

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 euclidean ring `ring`, which must contain all matrix entries. For compatibility reasons it can be omitted and defaults to the `DefaultRing`

(56.1-3) of the matrix entries.

The function `ElementaryDivisorsMatDestructive`

produces the same result but in the process may destroy the contents of `mat`.

gap> mat:=[[1,2,3],[4,5,6],[7,8,9]];; gap> ElementaryDivisorsMat(mat); [ 1, 3, 0 ] gap> x:=Indeterminate(Rationals,"x");; gap> mat:=mat*One(x)-x*mat^0; [ [ -x+1, 2, 3 ], [ 4, -x+5, 6 ], [ 7, 8, -x+9 ] ] gap> ElementaryDivisorsMat(PolynomialRing(Rationals,1),mat); [ 1, 1, x^3-15*x^2-18*x ] gap> mat:=KroneckerProduct(CompanionMat((x-1)^2), > CompanionMat((x^3-1)*(x-1)));; gap> mat:=mat*One(x)-x*mat^0; [ [ -x, 0, 0, 0, 0, 0, 0, 1 ], [ 0, -x, 0, 0, -1, 0, 0, -1 ], [ 0, 0, -x, 0, 0, -1, 0, 0 ], [ 0, 0, 0, -x, 0, 0, -1, -1 ], [ 0, 0, 0, -1, -x, 0, 0, -2 ], [ 1, 0, 0, 1, 2, -x, 0, 2 ], [ 0, 1, 0, 0, 0, 2, -x, 0 ], [ 0, 0, 1, 1, 0, 0, 2, -x+2 ] ] gap> ElementaryDivisorsMat(PolynomialRing(Rationals,1),mat); [ 1, 1, 1, 1, 1, 1, x-1, x^7-x^6-2*x^4+2*x^3+x-1 ]

`‣ ElementaryDivisorsTransformationsMat` ( [ring, ]mat ) | ( operation ) |

`‣ ElementaryDivisorsTransformationsMatDestructive` ( ring, mat ) | ( function ) |

`ElementaryDivisorsTransformations`

, in addition to the tasks done by `ElementaryDivisorsMat`

, also calculates transforming matrices. It returns a record with components `normal`

(a matrix \(S\)), `rowtrans`

(a matrix \(P\)), and `coltrans`

(a matrix \(Q\)) such that \(P A Q = S\). The operations are performed over the euclidean ring `ring`, which must contain all matrix entries. For compatibility reasons it can be omitted and defaults to the `DefaultRing`

(56.1-3) of the matrix entries.

The function `ElementaryDivisorsTransformationsMatDestructive`

produces the same result but in the process destroys the contents of `mat`.

gap> mat:=KroneckerProduct(CompanionMat((x-1)^2),CompanionMat((x^3-1)*(x-1)));; gap> mat:=mat*One(x)-x*mat^0; [ [ -x, 0, 0, 0, 0, 0, 0, 1 ], [ 0, -x, 0, 0, -1, 0, 0, -1 ], [ 0, 0, -x, 0, 0, -1, 0, 0 ], [ 0, 0, 0, -x, 0, 0, -1, -1 ], [ 0, 0, 0, -1, -x, 0, 0, -2 ], [ 1, 0, 0, 1, 2, -x, 0, 2 ], [ 0, 1, 0, 0, 0, 2, -x, 0 ], [ 0, 0, 1, 1, 0, 0, 2, -x+2 ] ] gap> t:=ElementaryDivisorsTransformationsMat(PolynomialRing(Rationals,1),mat); rec( coltrans := [ [ 0, 0, 0, 0, 0, 0, 1/6*x^2-7/9*x-1/18, -3*x^3-x^2-x-1 ], [ 0, 0, 0, 0, 0, 0, -1/6*x^2+x-1, 3*x^3-3*x^2 ], [ 0, 0, 0, 0, 0, 1, -1/18*x^4+1/3*x^3-1/3*x^2-1/9*x, x^5-x^4+2*x^2-2*x ], [ 0, 0, 0, 0, -1, 0, -1/9*x^3+1/2*x^2+1/9*x, 2*x^4+x^3+x^2+2*x ], [ 0, -1, 0, 0, 0, 0, -2/9*x^2+19/18*x, 4*x^3+x^2+x ], [ 0, 0, -1, 0, 0, -x, 1/18*x^5-1/3*x^4+1/3*x^3+1/9*x^2, -x^6+x^5-2*x^3+2*x^2 ], [ 0, 0, 0, -1, x, 0, 1/9*x^4-2/3*x^3+2/3*x^2+1/18*x, -2*x^5+2*x^4-x^2+x ], [ 1, 0, 0, 0, 0, 0, 1/6*x^3-7/9*x^2-1/18*x, -3*x^4-x^3-x^2-x ] ], normal := [ [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, x-1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, x^7-x^6-2*x^4+2*x^3+x-1 ] ], rowtrans := [ [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 1, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 1, 0, 0, 0, 0 ], [ -x+2, -x, 0, 0, 1, 0, 0, 0 ], [ 2*x^2-4*x+2, 2*x^2-x, 0, 2, -2*x+1, 0, 0, 1 ], [ 3*x^3-6*x^2+3*x, 3*x^3-2*x^2, 2, 3*x, -3*x^2+2*x, 0, 1, 2*x ], [ 1/6*x^8-7/6*x^7+2*x^6-4/3*x^5+7/3*x^4-4*x^3+13/6*x^2-7/6*x+2, 1/6*x^8-17/18*x^7+13/18*x^6-5/18*x^5+35/18*x^4-31/18*x^3+1/9*x^2-x+\ 2, 1/9*x^5-5/9*x^4+1/9*x^3-1/9*x^2+14/9*x-1/9, 1/6*x^6-5/6*x^5+1/6*x^4-1/6*x^3+11/6*x^2-1/6*x, -1/6*x^7+17/18*x^6-13/18*x^5+5/18*x^4-35/18*x^3+31/18*x^2-1/9*x+1, 1, 1/18*x^5-5/18*x^4+1/18*x^3-1/18*x^2+23/18*x-1/18, 1/9*x^6-5/9*x^5+1/9*x^4-1/9*x^3+14/9*x^2-1/9*x ] ] ) gap> t.rowtrans*mat*t.coltrans; [ [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, x-1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, x^7-x^6-2*x^4+2*x^3+x-1 ] ]

`‣ DiagonalizeMat` ( ring, mat ) | ( operation ) |

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 ] ]

`‣ SemiEchelonMat` ( mat ) | ( attribute ) |

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`

list of row vectors, each with pivot element the identity of \(F\),

`heads`

list that contains at position

`i`, if nonzero, the number of the row for that the pivot element is in column`i`.

`‣ SemiEchelonMatDestructive` ( mat ) | ( operation ) |

This does the same as `SemiEchelonMat( `

, except that it may (and probably will) destroy the matrix `mat` )`mat`.

gap> mm:=[[1,2,3],[4,5,6],[7,8,9]];; gap> SemiEchelonMatDestructive( mm ); rec( heads := [ 1, 2, 0 ], vectors := [ [ 1, 2, 3 ], [ 0, 1, 2 ] ] ) gap> mm; [ [ 1, 2, 3 ], [ 0, 1, 2 ], [ 0, 0, 0 ] ]

`‣ SemiEchelonMatTransformation` ( mat ) | ( attribute ) |

does the same as `SemiEchelonMat`

(24.10-1) but additionally stores the linear transformation \(T\) performed on the matrix. The additional components of the result are

`coeffs`

a list of coefficients vectors of the

`vectors`

component, with respect to the rows of`mat`, that is,`coeffs * mat`

is the`vectors`

component.`relations`

a list of basis vectors for the (left) null space of

`mat`.

gap> SemiEchelonMatTransformation([[1,2,3],[0,0,1]]); rec( coeffs := [ [ 1, 0 ], [ 0, 1 ] ], heads := [ 1, 0, 2 ], relations := [ ], vectors := [ [ 1, 2, 3 ], [ 0, 0, 1 ] ] )

`‣ SemiEchelonMats` ( mats ) | ( operation ) |

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`

(24.10-1)).

`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`

list of matrices, each with pivot element the identity of \(F\),

`heads`

matrix that contains at position [

`i`,`j`], if nonzero, the number of the matrix that has the pivot element in this position

`‣ SemiEchelonMatsDestructive` ( mats ) | ( operation ) |

Does the same as `SemiEchelonMats`

(24.10-4), except that it may destroy its argument. Therefore the argument must be a list of matrices that are mutable.

See also chapter 25

`‣ BaseMat` ( mat ) | ( attribute ) |

returns a basis for the row space generated by the rows of `mat` in the form of an immutable matrix.

`‣ BaseMatDestructive` ( mat ) | ( operation ) |

Does the same as `BaseMat`

(24.11-1), with the difference that it may destroy the matrix `mat`. The matrix `mat` must be mutable.

gap> mat:=[[1,2,3],[4,5,6],[7,8,9]];; gap> BaseMat(mat); [ [ 1, 2, 3 ], [ 0, 1, 2 ] ] gap> mm:= [[1,2,3],[4,5,6],[5,7,9]];; gap> BaseMatDestructive( mm ); [ [ 1, 2, 3 ], [ 0, 1, 2 ] ] gap> mm; [ [ 1, 2, 3 ], [ 0, 1, 2 ], [ 0, 0, 0 ] ]

`‣ BaseOrthogonalSpaceMat` ( mat ) | ( attribute ) |

Let \(V\) be the row space generated by the rows of `mat` (over any field that contains all entries of `mat`). `BaseOrthogonalSpaceMat( `

computes a base of the orthogonal space of \(V\).`mat` )

The rows of `mat` need not be linearly independent.

`‣ SumIntersectionMat` ( M1, M2 ) | ( operation ) |

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 24.10).

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 ) | ( function ) |

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`

s a basis of the space spanned by

`mat`in upper triangular form with leading ones at all echelon steps and zeroes above these ones.`factorspace`

is a list of extending vectors in upper triangular form.

`factorzero`

is a zero vector.

`heads`

is a list of integers which can be used to decompose vectors in the basis vectors. The

`i`th entry indicating the vector that gives an echelon step at position`i`. A negative number indicates an echelon step in the subspace, a positive number an echelon step in the complement, the absolute value gives the position of the vector in the lists`subspace`

and`factorspace`

.

gap> BaseSteinitzVectors(IdentityMat(3,1),[[11,13,15]]); rec( factorspace := [ [ 0, 1, 15/13 ], [ 0, 0, 1 ] ], factorzero := [ 0, 0, 0 ], heads := [ -1, 1, 2 ], subspace := [ [ 1, 13/11, 15/11 ] ] )

`‣ DiagonalOfMatrix` ( mat ) | ( function ) |

`‣ DiagonalOfMat` ( mat ) | ( function ) |

return the diagonal of the matrix `mat`. If `mat` is not a square matrix, then the result has the same length as the rows of `mat`, and is padded with zeros if `mat` has fewer rows than columns.

gap> DiagonalOfMatrix( [ [ 1, 2, 3 ], [ 4, 5, 6 ] ] ); [ 1, 5, 0 ]

`‣ UpperSubdiagonal` ( mat, pos ) | ( operation ) |

returns a mutable list containing the entries of the `pos`th upper subdiagonal of the matrix `mat`.

gap> UpperSubdiagonal( [ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ], 1 ); [ 2, 6 ] gap> UpperSubdiagonal( [ [ 1, 2 ], [ 3, 4 ], [ 5, 6 ] ], 1 ); [ 2 ] gap> UpperSubdiagonal( [ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8 ] ], 1 ); [ 2, 7 ]

`‣ DepthOfUpperTriangularMatrix` ( mat ) | ( attribute ) |

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

`‣ CharacteristicPolynomial` ( [F, E, ]mat[, ind] ) | ( attribute ) |

For a square matrix `mat`, `CharacteristicPolynomial`

returns the *characteristic polynomial* of `mat`, that is, the `StandardAssociate`

(56.5-5) of the determinant of the matrix \(\textit{mat} - X \cdot I\), where \(X\) is an indeterminate and \(I\) is the appropriate identity matrix.

If fields `F` and `E` are given, then `F` must be a subfield of `E`, and `mat` must have entries in `E`. Then `CharacteristicPolynomial`

returns the characteristic polynomial of the `F`-linear mapping induced by `mat` on the underlying `E`-vector space of `mat`. In this case, the characteristic polynomial is computed using `BlownUpMat`

(24.13-4) for the field extension of \(E/F\) generated by the default field. Thus, if \(F = E\), the result is the same as for the one argument version.

The returned polynomials are expressed in the indeterminate number `ind`. If `ind` is not given, it defaults to \(1\).

`CharacteristicPolynomial(`

is a multiple of the minimal polynomial `F`, `E`, `mat`)`MinimalPolynomial(`

(see `F`, `mat`)`MinimalPolynomial`

(66.8-1)).

Note that, up to **GAP** version 4.4.6, `CharacteristicPolynomial`

only allowed to specify one field (corresponding to `F`) as an argument. That usage has been disabled because its definition turned out to be ambiguous and may have lead to unexpected results. (To ensure backward compatibility, it is still possible to use the old form if `F` contains the default field of the matrix, see `DefaultFieldOfMatrix`

(24.4-2), but this feature will disappear in future versions of **GAP**.)

gap> CharacteristicPolynomial( [ [ 1, 1 ], [ 0, 1 ] ] ); x^2-2*x+1 gap> mat := [[0,1],[E(4)-1,E(4)]];; gap> CharacteristicPolynomial( mat ); x^2+(-E(4))*x+(1-E(4)) gap> CharacteristicPolynomial( Rationals, CF(4), mat ); x^4+3*x^2+2*x+2 gap> mat:= [ [ E(4), 1 ], [ 0, -E(4) ] ];; gap> CharacteristicPolynomial( mat ); x^2+1 gap> CharacteristicPolynomial( Rationals, CF(4), mat ); x^4+2*x^2+1

`‣ RationalCanonicalFormTransform` ( mat ) | ( function ) |

For a matrix `A`

, return a matrix `P`

such that \(A^{P}\) is in rational canonical form (also called Frobenius normal form). The algorithm used is the basic textbook version and thus not of optimal complexity.

gap> aa:=[[0,-8,12,40,-36,4,0,59,15,-9],[-2,-2,-2,6,-11,1,-1,10,1,0], > [1,5,0,-6,12,-2,0,-12,-4,2],[0,0,0,2,0,0,0,7,0,0], > [0,2,-3,-7,8,-1,0,-7,-3,2],[-5,-4,-6,18,-30,2,-2,35,5,-1], > [-1,-6,6,20,-28,3,0,24,10,-6],[0,0,0,-1,0,0,0,-3,0,0], > [0,0,-1,-2,-2,0,-1,-7,0,0],[0,-8,9,21,-36,4,-2,12,12,-8]];; gap> t:=RationalCanonicalFormTransform(aa);; gap> aa^t; [ [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 0, -1 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1, -1 ] ]

`‣ JordanDecomposition` ( mat ) | ( attribute ) |

`JordanDecomposition( `

returns a list `mat ` )`[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 ) | ( function ) |

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 \(F^n\) w.r.t. the \(F\)-basis with vectors \((v_1, ldots, v_n)\) and suppose that the basis `B` consists of the vectors \((b_1, \ldots, b_m)\); then the returned matrix is the matrix of the linear transformation on the row space \(K^{mn}\) w.r.t. the \(K\)-basis whose vectors are \((b_1 v_1, \ldots b_m v_1, \ldots, b_m v_n)\).

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 ) | ( function ) |

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`

(24.13-4) are compatible in the sense that for a matrix `mat` over \(F\), `BlownUpVector( `

is equal to `B`, `mat` * `vector` )`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

`‣ CompanionMatrix` ( poly ) | ( operation ) |

`‣ CompanionMat` ( poly ) | ( operation ) |

compute a companion matrix of the polynomial `poly`. This matrix has `poly` as its minimal polynomial.

Just as for row vectors, (see section 23.3), **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 able to be represented as a compact row vector of the same length 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> m := [v];; ConvertToMatrixRep(m,GF(2));; m; <a 1x5 matrix over GF2> gap> m := [v,v];; ConvertToMatrixRep(m,GF(2));; m; <a 2x5 matrix over GF2> gap> m := [v,v,v];; ConvertToMatrixRep(m,GF(2));; m; <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> m := [v];; ConvertToMatrixRep(m,GF(3));; m; [ [ 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];; ConvertToMatrixRep(m,GF(3));; m; < mutable compressed matrix 4x8 over GF(3) >

All compressed matrices over GF(2) are viewed as `<a `

, while over fields GF(q) for q between 3 and 256, matrices with 25 or more entries are viewed in this way, and smaller ones as lists of lists.`n`x`m` matrix over GF2>

Matrices can be converted to this special representation via the following functions.

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.

In order to form a row of a compressed matrix a vector must accept certain restrictions. Specifically, it cannot change its length or change the field over which it is compressed. The main consequences of this are: that only elements of the appropriate field can be assigned to entries of the vector, and only to positions between 1 and the original length; that the vector cannot be shared between two matrices compressed over different fields.

This is enforced by the filter `IsLockedRepresentationVector`

. When a vector becomes part of a compressed matrix, this filter is set for it. Assignment, `Unbind`

(21.5-3), `ConvertToVectorRep`

(23.3-1) and `ConvertToMatrixRep`

(24.14-2) are all prevented from altering a vector with this filter.

gap> v := [Z(2),Z(2)];; ConvertToVectorRep(v,GF(2));; v; <a GF2 vector of length 2> gap> m := [v,v]; [ <a GF2 vector of length 2>, <a GF2 vector of length 2> ] gap> ConvertToMatrixRep(m,GF(2)); 2 gap> m2 := [m[1], [Z(4),Z(4)]]; # now try and mix in some GF(4) [ <a GF2 vector of length 2>, [ Z(2^2), Z(2^2) ] ] gap> ConvertToMatrixRep(m2); # but m2[1] is locked #I ConvertToVectorRep: locked vector not converted to different field fail gap> m2 := [ShallowCopy(m[1]), [Z(4),Z(4)]]; # a fresh copy of row 1 [ <a GF2 vector of length 2>, [ Z(2^2), Z(2^2) ] ] 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" ]

Arithmetic operations (see 21.11 and the following sections) preserve the compression status of matrices in the sense that if all arguments are compressed matrices written over the same field and the result is a matrix then also the result is a compressed matrix written over this field.

There are also two operations that are only available for matrices written over finite fields.

`‣ ImmutableMatrix` ( field, matrix[, change] ) | ( operation ) |

Let `matrix` be an object for which either `IsMatrix`

(24.2-1) or `IsMatrixObj`

(26.2-2) returns `true`

. In the former case, `matrix` is a list of lists, and `ImmutableMatrix`

returns an immutable object for which `IsMatrix`

(24.2-1) returns `true`

(in particular again a list of lists), which is equal to `matrix`, and which is in the optimal (concerning space and runtime) representation for matrices defined over `field`, provided that the entries of `matrix` lie in `field`. In the latter case, `ImmutableMatrix`

returns an immutable object that is equal to the result of `ChangedBaseDomain`

(26.6-3) when this is called with `matrix` and `field`.

This means that matrices obtained by several calls of `ImmutableMatrix`

for the same `field` are compatible for fast arithmetic without need for field conversion.

If the input matrix `matrix` is in `IsMatrix`

(24.2-1) then it or its rows might change their representation as a side effect of this function. However, one cannot rely on this side effect. Also, if `matrix` is already immutable and the result of `ImmutableMatrix`

has the same internal representation as `matrix`, the result is not necessarily *identical* to `matrix`.

If `change` is `true`

, `matrix` or its rows (if there are subobjects that represent rows) may be changed to become immutable; otherwise the rows of `matrix` are copied first.

`‣ ConvertToMatrixRep` ( list[, field] ) | ( function ) |

`‣ ConvertToMatrixRep` ( list[, fieldsize] ) | ( function ) |

`‣ ConvertToMatrixRepNC` ( list[, field] ) | ( function ) |

`‣ ConvertToMatrixRepNC` ( list[, fieldsize] ) | ( function ) |

This function is more technical version of `ImmutableMatrix`

(24.14-1), which will never copy a matrix (or any rows of it) but may fail if it encounters rows locked in the wrong representation, or various other more technical problems. Most users should use `ImmutableMatrix`

(24.14-1) instead. The NC versions of the function do less checking of the argument and may cause unpredictable results or crashes if given unsuitable arguments. Called with one argument `list`, `ConvertToMatrixRep`

converts `list` to an internal matrix representation if possible.

Called with a list `list` and a finite field `field`, `ConvertToMatrixRep`

converts `list` to an internal matrix representation appropriate for a matrix over `field`.

Instead of a `field` also its size `fieldsize` may be given.

It is forbidden to call this function unless all elements of `list` are row vectors with entries in the field `field`. Violation of this condition can lead to unpredictable behaviour or a system crash. (Setting the assertion level to at least 2 might catch some violations before a crash, see `SetAssertionLevel`

(7.5-1).)

`list` may already be a compressed matrix. In this case, if no `field` or `fieldsize` is given, then nothing happens.

The return value is the size of the field over which the matrix ends up written, if it is written in a compressed representation.

`‣ ProjectiveOrder` ( mat ) | ( attribute ) |

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 ) | ( function ) |

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(`

for some `q`^`r`)`r` are powers of an element \(\xi\) in the splitting field, which is of order `expo`. `SimultaneousEigenvalues`

returns a matrix of integers mod `expo` \((a_{{i,j}})\), such that the power \(\xi^{{a_{{i,j}}}}\) is an eigenvalue of the `i`-th matrix in `matlist` and the eigenspaces of the different matrices to the eigenvalues \(\xi^{{a_{{i,j}}}}\) for fixed `j` are equal.

The following operations 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 \(p\), this is effectively computation in a matrix over the prime field in characteristic \(p\).

`‣ InverseMatMod` ( mat, obj ) | ( operation ) |

For a square matrix `mat`, `InverseMatMod`

returns a matrix `inv` such that

is congruent to the identity matrix modulo `inv` * `mat``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 ] ]

`‣ BasisNullspaceModN` ( M, n ) | ( function ) |

`M` must be a matrix of integers and `n` a positive integer. Then `BasisNullspaceModN`

returns a set `B` of vectors such that every vector `v` of integer modulo `n` satisfying `v` `M` = 0 modulo `n` can be expressed by a Z-linear combination of elements of `B`.

`‣ NullspaceModQ` ( M, q ) | ( function ) |

`‣ NullspaceModN` ( M, n ) | ( function ) |

`M` must be a matrix of integers and `n` a positive integer. Then `NullspaceModN`

returns the set of all vectors of integers modulo `n`, which solve the homogeneous equation system `v` `M` = 0 modulo `n`.

`NullspaceModQ`

is a synonym for `NullspaceModN`

.

gap> NullspaceModN( [ [ 2 ] ], 8 ); [ [ 0 ], [ 4 ] ] gap> NullspaceModN( [ [ 2, 1 ], [ 0, 2 ] ], 6 ); [ [ 0, 0 ], [ 0, 3 ] ] gap> mat:= [ [ 1, 3 ], [ 1, 2 ], [ 1, 1 ] ];; gap> NullspaceModN( mat, 5 ); [ [ 0, 0, 0 ], [ 1, 3, 1 ], [ 2, 1, 2 ], [ 3, 4, 3 ], [ 4, 2, 4 ] ]

When multiplying two compressed matrices \(M\) and \(N\) over GF(2) of dimensions \(a \times b\) and \(b \times c\), where \(a\), \(b\) and \(c\) are all greater than or equal to 128, **GAP** by default uses a more sophisticated matrix multiplication algorithm, in which linear combinations of groups of 8 rows of \(M\) are remembered and re-used in constructing various rows of the product. This is called level 8 grease. To optimise memory access patterns, these combinations are stored for \((b+255)/256\) sets of 8 rows at once. This number is called the blocking level.

These levels of grease and blocking are found experimentally to give good performance across a range of processors and matrix sizes, but other levels may do even better in some cases. You can control the levels exactly using the functions below.

We plan to include greased blocked matrix multiplication for other finite fields, and greased blocked algorithms for inversion and other matrix operations in a future release.

`‣ PROD_GF2MAT_GF2MAT_SIMPLE` ( m1, m2 ) | ( function ) |

This function performs the standard unblocked and ungreased matrix multiplication for matrices of any size.

`‣ PROD_GF2MAT_GF2MAT_ADVANCED` ( m1, m2, g, b ) | ( function ) |

This function computes the product of `m1` and `m2`, which must be compressed matrices over GF(2) of compatible dimensions, using level `g` grease and level `b` blocking.

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 ) | ( function ) |

returns a block matrix with `nrb` row blocks and `ncb` column blocks which is equal to the ordinary matrix `m`.

`‣ BlockMatrix` ( blocks, nrb, ncb[, rpb, cpb, zero] ) | ( function ) |

`BlockMatrix`

returns an immutable matrix in the sparse representation `IsBlockMatrixRep`

. The nonzero blocks are described by the list `blocks` of triples \([ \textit{i}, \textit{j}, M(i,j) ]\) each consisting of a matrix \(M(i,j)\) and its block coordinates in the block matrix to be constructed. All matrices \(M(i,j)\) must have the same dimensions. As usual the first coordinate specifies the row and the second one the column. The resulting 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`

(24.2-2)), and also the block matrix is an ordinary matrix.

gap> M := BlockMatrix([[1,1,[[1, 2],[ 3, 4]]], > [1,2,[[9,10],[11,12]]], > [2,2,[[5, 6],[ 7, 8]]]],2,2); <block matrix of dimensions (2*2)x(2*2)> gap> Display(M); [ [ 1, 2, 9, 10 ], [ 3, 4, 11, 12 ], [ 0, 0, 5, 6 ], [ 0, 0, 7, 8 ] ]

`‣ MatrixByBlockMatrix` ( blockmat ) | ( attribute ) |

returns a plain ordinary matrix that is equal to the block matrix `blockmat`.

`‣ SimplexMethod` ( A, b, c ) | ( function ) |

Find a rational vector `x` that maximizes \(\textit{x}\cdot\textit{c}\), subject to the constraint \(\textit{A}\cdot\textit{x}\le\textit{b}\).

gap> A:=[[3,1,1,4],[1,-3,2,3],[2,1,3,-1]];; gap> b:=[12,7,10];;c:=[2,4,3,1];; gap> SimplexMethod(A,b,c); [ [ 0, 52/5, 0, 2/5 ], 42 ]

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