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# 24 Matrices

### Sections

1. Categories of Matrices
2. Operators for Matrices
3. Properties and Attributes of Matrices
4. Matrix Constructions
5. Random Matrices
6. Matrices Representing Linear Equations and the Gaussian Algorithm
7. Eigenvectors and eigenvalues
8. Elementary Divisors
9. Echelonized Matrices
10. Matrices as Basis of a Row Space
11. Triangular Matrices
12. Matrices as Linear Mappings
13. Matrices over Finite Fields
14. Special Multiplication Algorithms for Matrices over GF(2)
15. Block Matrices

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. 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 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).

Note that, since a matrix is a list of lists, the behaviour of `ShallowCopy` for matrices is just a special case of `ShallowCopy` for lists (see Duplication of Lists); called with an immutable matrix mat, `ShallowCopy` 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 V`

  The info class for matrix operations is `InfoMatrix`.

  ## 24.1 Categories of Matrices

- `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`.

  ## 24.2 Operators for Matrices

  The rules for arithmetic operations involving matrices are in fact special cases of those for the arithmetic of lists, given in Section Arithmetic for Lists 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`, and the multiplicative behaviour is defined only for lists in the category `IsMultiplicativeGeneralizedRowVector` (see Filters Controlling the Arithmetic Behaviour of Lists).

- ``mat1`+` mat2 `O`

  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 `O`
- ``mat`+` scalar `O`

  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 `O`
- ``mat`-` scalar `O`

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

- ``scalar`*` mat `O`
- ``mat`*` scalar `O`

  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 `O`

  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 the same length m, say; 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 `O`

  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 `O`

  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, say, 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 mat1 [i][l] * mat2 [l][j], with l running from 1 to n.

- `Inverse(` mat`) O`

  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 `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 invertible.

- ``mat`^` int `O`
- ``mat1`^` mat2 `O`
- ``vec`^` mat `O`

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

- `Comm(` mat1`,` mat2`) O`

  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 Arithmetic for Lists.

- ``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 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 `O`
- ``matlist`*` mat `O`

  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 `O`

  Dividing a list matlist of matrices by an invertible matrix mat evaluates to ``matlist`*` mat`^-1`.

- ``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.

  For the mutability of results of arithmetic operations, see Mutability and Copyability.

  ## 24.3 Properties and Attributes of Matrices

- `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`

  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 Creating Finite Fields and Operations for Cyclotomics).

      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.

  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.

- `DeterminantMatDestructive(` mat`) O`

  Does the same as `DeterminantMat`, with the difference that it may destroy its argument. The matrix mat must be mutable.

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

- `DeterminantMatDivFree(` mat`) O`

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

- `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.

  ## 24.4 Matrix Constructions

- `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 ] ]

- `TransposedMatImmutable(` mat`) A`
- `TransposedMatAttr(` mat`) AM`
- `TransposedMat(` mat`) AM`
- `TransposedMatMutable(` mat`) O`
- `TransposedMatOp(` mat`) O`

  These functions all return the transposed of the matrix mat, i.e., a matrix trans such that ``trans`[`i`][`k`] =` 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` (see Zero).

- `TransposedMatDestructive(` mat`) O`

  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 ] ]

- `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 Basis), 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 E).

  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

  M = In +

  vtr

  ·

  w−1

  v

  vtr

  ·v

  where `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.

- `MutableCopyMat(` mat`) O`

  `MutableCopyMat` returns a fully mutable copy of the matrix mat.

  The default method does `List(`mat`,ShallowCopy)` and thus may also be called for the empty list, returning a new empty list.

  ## 24.5 Random Matrices

- `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));
      [ [ 0*Z(3), Z(3), Z(3)^0 ], [ Z(3), 0*Z(3), Z(3)^0 ] ]
      gap> RandomInvertibleMat(4);
      [ [ 4, 3, 5, 3 ], [ 1, -2, 1, -1 ], [ -2, 4, -1, -3 ], [ -1, 1, 2, 0 ] ]

  ## 24.6 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:=TransposedMatMutable(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.

- `NullspaceMatDestructive(` mat`) O`
- `TriangulizedNullspaceMatDestructive(` mat`) O`

  This function does the same as `NullspaceMat`. However, the latter function makes a copy of mat to avoid having to change it. This function does not do that; it returns the null space 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`) O`

  returns a row vector x that is a solution of the equation ``x`*` mat`=` vec``. It returns `fail` if no such vector exists.

- `SolutionMatDestructive(` mat`,` vec`) O`

  Does the same as `SolutionMat(` mat`,` vec`)` except 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> SolutionMat(mat,[3,5,7]);
      [ 5/3, 1/3, 0 ]
      gap> mm:=[[1,2,3],[4,5,6],[7,8,9]];;
      gap> SolutionMatDestructive( mm, [3,5,7] );
      [ 5/3, 1/3, 0 ]
      gap> mm;
      [ [ 1, 2, 3 ], [ 0, -3, -6 ], [ 0, 0, 0 ] ]

- `BaseFixedSpace(` mats`) F`

  `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 ] ]

  ## 24.7 Eigenvectors and eigenvalues

- `GeneralisedEigenvalues(` F`,` A`) O`
- `GeneralizedEigenvalues(` F`,` A`) O`

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

- `GeneralisedEigenspaces(` F`,` A`) O`
- `GeneralizedEigenspaces(` F`,` A`) O`

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

- `Eigenvalues(` F`,` A`) O`

  The eigenvalues of the matrix A over the field F.

- `Eigenspaces(` F`,` A`) O`

  The eigenspaces of the matrix A over the field F.

- `Eigenvectors(` F`,` A`) O`

  The eigenspaces of the matrix A over the field F.

  ## 24.8 Elementary Divisors

  See also chapter Integral Matrices and Lattices.

- `ElementaryDivisorsMat( [`ring`, ]` mat`) O`
- `ElementaryDivisorsMatDestructive(` ring`,` mat`) F`

  `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`.

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

      gap> mat:=[[1,2,3],[4,5,6],[7,8,9]];;
      gap> ElementaryDivisorsMat(mat);
      [ 1, 3, 0 ]
      gap> x:=X(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 ]

- `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 ] ]

  ## 24.9 Echelonized Matrices

- `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`
    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`) O`

  This does the same as `SemiEchelonMat(` mat`)`, except that it may (and probably will) destroy the matrix 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`) 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`
    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( heads := [ 1, 0, 2 ], vectors := [ [ 1, 2, 3 ], [ 0, 0, 1 ] ],
        coeffs := [ [ 1, 0 ], [ 0, 1 ] ], relations := [  ] )

- `SemiEchelonMats(` mats`) O`

  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`
    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`) O`

  Does the same as `SemiEchelonmats`, except that it may destroy its argument. Therefore the argument must be a list of matrices that re mutable.

  ## 24.10 Matrices as Basis of a Row Space

- `BaseMat(` mat`) A`

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

- `BaseMatDestructive(` mat`) O`

  Does the same as `BaseMat`, 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`) 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`
    is 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 ith 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 ], subspace := [ [ 1, 13/11, 15/11 ] ],
        heads := [ -1, 1, 2 ] )

  See also chapter Integral Matrices and Lattices

  ## 24.11 Triangular Matrices

- `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

  ## 24.12 Matrices as Linear Mappings

- `CharacteristicPolynomial(` mat`) A`
- `CharacteristicPolynomial( [`F`, ]`mat`[,` ind`] ) O`

  For a square matrix mat, `CharacteristicPolynomial` returns the characteristic polynomial of mat, that is, the `StandardAssociate` of the determinant of the matrix mat − X ·I, where X is an indeterminate and I is the appropriate identity matrix.

  If a field F is given as first argument then the characteristic polynomial of the F-linear mapping induced by mat is computed. If F contains the entries of mat then this is of course the same polynomial as the one computed by the one argument version; if F is a proper subfield of the default field (see DefaultFieldOfMatrix) of mat then the characteristic polynomial is computed using `BlownUpMat` (see BlownUpMat).

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

  The characteristic polynomial is a multiple of the minimal polynomial (see MinimalPolynomial).

      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, 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, mat );
      x^4+2*x^2+1

- `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, …, 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, …bm v1, …, 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.

  ## 24.13 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 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` n `x` m `matrix over GF2>`, 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.

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

- `ImmutableMatrix(` field`,` matrix`, [`change`] ) F`

  returns an immutable matrix equal to matrix which is in the most compact representation possible over field. The input matrix matrix or its rows might change the representation, however the result of `ConvertedMatrix` is not necessarily identical to matrix if a conversion is not possible. If change is `true`, the rows of `matrix` (or `matrix` itself) may be changed to become immutable (otherwise they are copied first).

- `ConvertToMatrixRep(` list`) F`
- `ConvertToMatrixRep(` list`,` field`) F`
- `ConvertToMatrixRep(` list`,` fieldsize`) F`
- `ConvertToMatrixRepNC(` list`) F`
- `ConvertToMatrixRepNC(` list`,` field`) F`
- `ConvertToMatrixRepNC(` 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. 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.)

  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.

  In general, it is better to call `ImmutableMatrix` (see ImmutableMatrix) instead since this function can also deal with mutable rows or rows locked in a wrong representation.

  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`, `ConvertToVectorRep` and `ConvertToMatrixRep` 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 Arithmetic for Lists 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.

- `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 ξ in the splitting field, which is of order expo. `SimultaneousEigenvalues` returns a matrix of integers mod expo, say (ai,j), such that the power ξai,j is an eigenvalue of the i-th matrix in matlist and the eigenspaces of the different matrices to the eigenvalues ξai,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 p, say, this is effectively computation in a matrix over the prime field in characteristic p.

- `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 ] ]

  ## 24.14 Special Multiplication Algorithms for Matrices over GF(2)

  When multiplying two compressed matrices M and N over GF(2) of dimensions a×b and b×c, say, 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:

- `PROD_GF2MAT_GF2MAT_SIMPLE(` m1`,` m2`) F`

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

- `PROD_GF2MAT_GF2MAT_ADVANCED(` m1`,` m2`,` g`,` b`) F`

  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.

  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.

  ## 24.15 Block Matrices

  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.

      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`) A`

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

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  GAP 4 manual
  August 2005