  
  [1X24 [33X[0;0YMatrices[133X[101X
  
  [33X[0;0YIn  [5XGAP[105X, Matrices can be represented by lists of row vectors, see [14X23[114X. (For a
  more general way to represent vectors and matrices, see Chapter [14X26[114X). 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.[133X
  
  [33X[0;0YBecause  matrices  are  just  a  special  case  of lists, all operations and
  functions  for  lists are applicable to matrices also (see chapter [14X21[114X). This
  especially  includes  accessing  elements  of  a matrix (see [14X21.3[114X), changing
  elements of a matrix (see [14X21.4[114X), and comparing matrices (see [14X21.10[114X).[133X
  
  [33X[0;0YNote  that,  since a matrix is a list of lists, the behaviour of [2XShallowCopy[102X
  ([14X12.7-1[114X)  for  matrices  is  just a special case of [2XShallowCopy[102X ([14X12.7-1[114X) for
  lists  (see [14X21.7[114X); called with an immutable matrix [3Xmat[103X, [2XShallowCopy[102X ([14X12.7-1[114X)
  returns  a  mutable  matrix  whose rows are identical to the rows of [3Xmat[103X. In
  particular  the  rows  are  still  immutable. To get a matrix whose rows are
  mutable, one can use [10XList( [3Xmat[103X[10X, ShallowCopy )[110X.[133X
  
  
  [1X24.1 [33X[0;0YInfoMatrix (Info Class)[133X[101X
  
  [1X24.1-1 InfoMatrix[101X
  
  [33X[1;0Y[29X[2XInfoMatrix[102X [32X info class[133X
  
  [33X[0;0YThe info class for matrix operations is [2XInfoMatrix[102X.[133X
  
  
  [1X24.2 [33X[0;0YCategories of Matrices[133X[101X
  
  [1X24.2-1 IsMatrix[101X
  
  [33X[1;0Y[29X[2XIsMatrix[102X( [3Xobj[103X ) [32X Category[133X
  
  [33X[0;0YBy convention [13Xmatrix[113X is a list of lists of equal length whose entries lie in
  a common ring.[133X
  
  [33X[0;0YFor technical reasons laid out at the top of Chapter [14X24[114X, the filter [2XIsMatrix[102X
  is  a  synonym  for  a  table  of  ring  elements, (see [2XIsTable[102X ([14X21.1-4[114X) and
  [2XIsRingElement[102X  ([14X31.14-16[114X)). This means that [2XIsMatrix[102X returns [9Xtrue[109X for tables
  such  as  [10X[[1,2],[3]][110X. If necessary, [2XIsRectangularTable[102X ([14X21.1-5[114X) can be used
  to  test  whether  an  object is a list of homogenous lists of equal lengths
  manually.[133X
  
  [33X[0;0YNote  that  matrices  may  have different multiplications, besides the usual
  matrix product there is for example the Lie product. So there are categories
  such as [2XIsOrdinaryMatrix[102X ([14X24.2-2[114X) and [2XIsLieMatrix[102X ([14X24.2-3[114X) that describe the
  matrix multiplication. One can form the product of two matrices only if they
  support the same multiplication.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat:=[[1,2,3],[4,5,6],[7,8,9]];[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ][128X[104X
    [4X[25Xgap>[125X [27XIsMatrix(mat);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xmat:=[[1,2],[3]];[127X[104X
    [4X[28X[ [ 1, 2 ], [ 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XIsMatrix(mat);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsRectangularTable(mat);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  the empty list [10X[][110X and more complex [21Xempty[121X structures such as [10X[[]][110X
  are  [13Xnot[113X  matrices,  although special methods allow them be used in place of
  matrices in some situations. See [2XEmptyMatrix[102X ([14X24.5-3[114X) below.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X[[0]]*[[]];[127X[104X
    [4X[28X[ [  ] ][128X[104X
    [4X[25Xgap>[125X [27XIsMatrix([[]]);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X24.2-2 IsOrdinaryMatrix[101X
  
  [33X[1;0Y[29X[2XIsOrdinaryMatrix[102X( [3Xobj[103X ) [32X Category[133X
  
  [33X[0;0YAn  [13Xordinary  matrix[113X is a matrix whose multiplication is the ordinary matrix
  multiplication.[133X
  
  [33X[0;0YEach  matrix in internal representation is in the category [2XIsOrdinaryMatrix[102X,
  and  arithmetic  operations  with  objects in [2XIsOrdinaryMatrix[102X produce again
  matrices in [2XIsOrdinaryMatrix[102X.[133X
  
  [33X[0;0YNote  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.[133X
  
  [1X24.2-3 IsLieMatrix[101X
  
  [33X[1;0Y[29X[2XIsLieMatrix[102X( [3Xmat[103X ) [32X Category[133X
  
  [33X[0;0YA  [13XLie  matrix[113X is a matrix whose multiplication is given by the Lie bracket.
  (Note  that  a matrix with ordinary matrix multiplication is in the category
  [2XIsOrdinaryMatrix[102X ([14X24.2-2[114X).)[133X
  
  [33X[0;0YEach  matrix  created  by [2XLieObject[102X ([14X64.1-1[114X) is in the category [2XIsLieMatrix[102X,
  and arithmetic operations with objects in [2XIsLieMatrix[102X produce again matrices
  in [2XIsLieMatrix[102X.[133X
  
  
  [1X24.3 [33X[0;0YOperators for Matrices[133X[101X
  
  [33X[0;0YThe  rules  for arithmetic operations involving matrices are in fact special
  cases  of  those for the arithmetic of lists, given in Section [14X21.11[114X and the
  following  sections,  here  we reiterate that definition, in the language of
  vectors and matrices.[133X
  
  [33X[0;0YNote that the additive behaviour sketched below is defined only for lists in
  the   category  [2XIsGeneralizedRowVector[102X  ([14X21.12-1[114X),  and  the  multiplicative
  behaviour     is    defined    only    for    lists    in    the    category
  [2XIsMultiplicativeGeneralizedRowVector[102X ([14X21.12-2[114X) (see [14X21.12[114X).[133X
  
  [33X[0;0Y[10X[3Xmat1[103X[10X + [3Xmat2[103X[10X[110X[133X
  
  [33X[0;0Yreturns  the  sum of the two matrices [3Xmat1[103X and [3Xmat2[103X, Probably the most usual
  situation  is  that  [3Xmat1[103X  and [3Xmat2[103X 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.[133X
  
  [33X[0;0YIn  more  general  situations, the sum of two matrices need not be a matrix,
  for  example  adding  an integer matrix [3Xmat1[103X and a matrix [3Xmat2[103X over a finite
  field  yields the table of pointwise sums, which will be a mixture of finite
  field elements and integers if [3Xmat1[103X has bigger dimensions than [3Xmat2[103X.[133X
  
  [33X[0;0Y[10X[3Xscalar[103X[10X + [3Xmat[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xmat[103X[10X + [3Xscalar[103X[10X[110X[133X
  
  [33X[0;0Yreturns  the  sum of the scalar [3Xscalar[103X and the matrix [3Xmat[103X. Probably the most
  usual  situation  is  that  the  entries  of  [3Xmat[103X lie in a common field with
  [3Xscalar[103X;  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.[133X
  
  [33X[0;0YMore  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.[133X
  
  [33X[0;0Y[10X[3Xmat1[103X[10X - [3Xmat2[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xscalar[103X[10X - [3Xmat[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xmat[103X[10X - [3Xscalar[103X[10X[110X[133X
  
  [33X[0;0YSubtracting a matrix or scalar is defined as adding its additive inverse, so
  the statements for the addition hold likewise.[133X
  
  [33X[0;0Y[10X[3Xscalar[103X[10X * [3Xmat[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xmat[103X[10X * [3Xscalar[103X[10X[110X[133X
  
  [33X[0;0Yreturns  the  product  of the scalar [3Xscalar[103X and the matrix [3Xmat[103X. Probably the
  most  usual situation is that the elements of [3Xmat[103X lie in a common field with
  [3Xscalar[103X;  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.[133X
  
  [33X[0;0YMore 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.[133X
  
  [33X[0;0Y[10X[3Xvec[103X[10X * [3Xmat[103X[10X[110X[133X
  
  [33X[0;0Yreturns  the  product of the row vector [3Xvec[103X and the matrix [3Xmat[103X. Probably the
  most  usual  situation  is  that  [3Xvec[103X  and [3Xmat[103X have the same lengths and are
  defined  over  a  common  field,  and  that all rows of [3Xmat[103X have some common
  length  [22Xm[122X; in this case the product is a new row vector of length [22Xm[122X over the
  same  field which is the sum of the scalar multiples of the rows of [3Xmat[103X with
  the corresponding entries of [3Xvec[103X.[133X
  
  [33X[0;0YMore 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.[133X
  
  [33X[0;0Y[10X[3Xmat[103X[10X * [3Xvec[103X[10X[110X[133X
  
  [33X[0;0Yreturns  the  product of the matrix [3Xmat[103X and the row vector [3Xvec[103X. (This is the
  standard  product of a matrix with a [13Xcolumn[113X vector.) Probably the most usual
  situation  is  that  the length of [3Xvec[103X and of all rows of [3Xmat[103X are equal, and
  that  the  elements  of  [3Xmat[103X and [3Xvec[103X lie in a common field; in this case the
  product  is  a  new  row  vector of the same length as [3Xmat[103X and over the same
  field  which  is  the sum of the scalar multiples of the columns of [3Xmat[103X with
  the corresponding entries of [3Xvec[103X.[133X
  
  [33X[0;0YMore 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.[133X
  
  [33X[0;0Y[10X[3Xmat1[103X[10X * [3Xmat2[103X[10X[110X[133X
  
  [33X[0;0YThis  form  evaluates  to  the (Cauchy) product of the two matrices [3Xmat1[103X and
  [3Xmat2[103X.  Probably  the  most  usual situation is that the number of columns of
  [3Xmat1[103X equals the number of rows of [3Xmat2[103X, and that the elements of [3Xmat[103X and [3Xvec[103X
  lie  in  a  common  field; if [3Xmat1[103X is a matrix with [22Xm[122X rows and [22Xn[122X columns and
  [3Xmat2[103X  is a matrix with [22Xn[122X rows and [22Xo[122X columns, the result is a new matrix with
  [22Xm[122X  rows  and [22Xo[122X columns. The element in row [22Xi[122X at position [22Xj[122X of the product is
  the sum of [22X[3Xmat1[103X[i][l] * [3Xmat2[103X[l][j][122X, with [22Xl[122X running from [22X1[122X to [22Xn[122X.[133X
  
  [33X[0;0Y[10XInverse( [3Xmat[103X[10X )[110X[133X
  
  [33X[0;0Yreturns  the  inverse  of the matrix [3Xmat[103X, which must be an invertible square
  matrix. If [3Xmat[103X is not invertible then [9Xfail[109X is returned.[133X
  
  [33X[0;0Y[10X[3Xmat1[103X[10X / [3Xmat2[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xscalar[103X[10X / [3Xmat[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xmat[103X[10X / [3Xscalar[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xvec[103X[10X / [3Xmat[103X[10X[110X[133X
  
  [33X[0;0YIn  general,  [10X[3Xleft[103X[10X  / [3Xright[103X[10X[110X is defined as [10X[3Xleft[103X[10X * [3Xright[103X[10X^-1[110X. Thus in the above
  forms the right operand must always be invertible.[133X
  
  [33X[0;0Y[10X[3Xmat[103X[10X ^ [3Xint[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xmat1[103X[10X ^ [3Xmat2[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xvec[103X[10X ^ [3Xmat[103X[10X[110X[133X
  
  [33X[0;0YPowering  a  square  matrix [3Xmat[103X by an integer [3Xint[103X yields the [3Xint[103X-th power of
  [3Xmat[103X;  if  [3Xint[103X  is negative then [3Xmat[103X must be invertible, if [3Xint[103X is [10X0[110X then the
  result is the identity matrix [10XOne( [3Xmat[103X[10X )[110X, even if [3Xmat[103X is not invertible.[133X
  
  [33X[0;0YPowering  a  square  matrix  [3Xmat1[103X by an invertible square matrix [3Xmat2[103X of the
  same  dimensions  yields  the  conjugate  of  [3Xmat1[103X by [3Xmat2[103X, i.e., the matrix
  [10X[3Xmat2[103X[10X^-1 * [3Xmat1[103X[10X * [3Xmat2[103X[10X[110X.[133X
  
  [33X[0;0YPowering  a row vector [3Xvec[103X by a matrix [3Xmat[103X is in every respect equivalent to
  [10X[3Xvec[103X[10X  * [3Xmat[103X[10X[110X. This operations reflects the fact that matrices act naturally on
  row vectors by multiplication from the right, and that the powering operator
  is [5XGAP[105X's standard for group actions.[133X
  
  [33X[0;0Y[10XComm( [3Xmat1[103X[10X, [3Xmat2[103X[10X )[110X[133X
  
  [33X[0;0Yreturns  the  commutator  of the square invertible matrices [3Xmat1[103X and [3Xmat2[103X of
  the  same  dimensions and over a common field, which is the matrix [10X[3Xmat1[103X[10X^-1 *
  [3Xmat2[103X[10X^-1 * [3Xmat1[103X[10X * [3Xmat2[103X[10X[110X.[133X
  
  [33X[0;0YThe  following  cases are still special cases of the general list arithmetic
  defined in [14X21.11[114X.[133X
  
  [33X[0;0Y[10X[3Xscalar[103X[10X + [3Xmatlist[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xmatlist[103X[10X + [3Xscalar[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xscalar[103X[10X - [3Xmatlist[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xmatlist[103X[10X - [3Xscalar[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xscalar[103X[10X * [3Xmatlist[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xmatlist[103X[10X * [3Xscalar[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xmatlist[103X[10X / [3Xscalar[103X[10X[110X[133X
  
  [33X[0;0YA  scalar  [3Xscalar[103X may also be added, subtracted, multiplied with, or divided
  into  a list [3Xmatlist[103X 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 [3Xmatlist[103X.[133X
  
  [33X[0;0Y[10X[3Xmat[103X[10X * [3Xmatlist[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xmatlist[103X[10X * [3Xmat[103X[10X[110X[133X
  
  [33X[0;0YA  matrix  [3Xmat[103X  may  also be multiplied with a list [3Xmatlist[103X of matrices. The
  result is a new list of matrices, where each entry is the product of [3Xmat[103X and
  the corresponding entry in [3Xmatlist[103X.[133X
  
  [33X[0;0Y[10X[3Xmatlist[103X[10X / [3Xmat[103X[10X[110X[133X
  
  [33X[0;0YDividing a list [3Xmatlist[103X of matrices by an invertible matrix [3Xmat[103X evaluates to
  [10X[3Xmatlist[103X[10X * [3Xmat[103X[10X^-1[110X.[133X
  
  [33X[0;0Y[10X[3Xvec[103X[10X * [3Xmatlist[103X[10X[110X[133X
  
  [33X[0;0Yreturns  the  product  of  the  vector [3Xvec[103X and the list of matrices [3Xmat[103X. The
  lengths  [3Xl[103X  of  [3Xvec[103X  and [3Xmatlist[103X must be equal. All matrices in [3Xmatlist[103X must
  have  the  same  dimensions.  The  elements  of  [3Xvec[103X and the elements of the
  matrices  in  [3Xmatlist[103X must lie in a common ring. The product is the sum over
  [10X[3Xvec[103X[10X[[3Xi[103X[10X] * [3Xmatlist[103X[10X[[3Xi[103X[10X][110X with [3Xi[103X running from 1 to [3Xl[103X.[133X
  
  [33X[0;0YFor the mutability of results of arithmetic operations, see [14X12.6[114X.[133X
  
  
  [1X24.4 [33X[0;0YProperties and Attributes of Matrices[133X[101X
  
  [1X24.4-1 DimensionsMat[101X
  
  [33X[1;0Y[29X[2XDimensionsMat[102X( [3Xmat[103X ) [32X attribute[133X
  
  [33X[0;0Yis  a list of length 2, the first being the number of rows, the second being
  the number of columns of the matrix [3Xmat[103X. If [3Xmat[103X is malformed, that is, it is
  not a [2XIsRectangularTable[102X ([14X21.1-5[114X), returns [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDimensionsMat([[1,2,3],[4,5,6]]);[127X[104X
    [4X[28X[ 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27XDimensionsMat([[1,2,3],[4,5]]);[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [1X24.4-2 DefaultFieldOfMatrix[101X
  
  [33X[1;0Y[29X[2XDefaultFieldOfMatrix[102X( [3Xmat[103X ) [32X attribute[133X
  
  [33X[0;0YFor   a  matrix  [3Xmat[103X,  [2XDefaultFieldOfMatrix[102X  returns  either  a  field  (not
  necessarily the smallest one) containing all entries of [3Xmat[103X, or [9Xfail[109X.[133X
  
  [33X[0;0YIf  [3Xmat[103X  is  a  matrix  of finite field elements or a matrix of cyclotomics,
  [2XDefaultFieldOfMatrix[102X  returns  the  default  field  generated  by the matrix
  entries (see [14X59.3[114X and [14X18.1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDefaultFieldOfMatrix([[Z(4),Z(8)]]);[127X[104X
    [4X[28XGF(2^6)[128X[104X
  [4X[32X[104X
  
  [1X24.4-3 TraceMatrix[101X
  
  [33X[1;0Y[29X[2XTraceMatrix[102X( [3Xmat[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XTraceMat[102X( [3Xmat[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XTrace[102X( [3Xmat[103X ) [32X attribute[133X
  
  [33X[0;0YThe trace of a square matrix is the sum of its diagonal entries.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XTraceMatrix([[1,2,3],[4,5,6],[7,8,9]]);[127X[104X
    [4X[28X15[128X[104X
  [4X[32X[104X
  
  [1X24.4-4 DeterminantMatrix[101X
  
  [33X[1;0Y[29X[2XDeterminantMatrix[102X( [3Xmat[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XDeterminantMat[102X( [3Xmat[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XDeterminant[102X( [3Xmat[103X ) [32X operation[133X
  
  [33X[0;0Yreturns the determinant of the square matrix [3Xmat[103X.[133X
  
  [33X[0;0YThese  methods assume implicitly that [3Xmat[103X is defined over an integral domain
  whose  quotient  field  is  implemented in [5XGAP[105X. For matrices defined over an
  arbitrary commutative ring with one see [2XDeterminantMatDivFree[102X ([14X24.4-6[114X).[133X
  
  [1X24.4-5 DeterminantMatrixDestructive[101X
  
  [33X[1;0Y[29X[2XDeterminantMatrixDestructive[102X( [3Xmat[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDeterminantMatDestructive[102X( [3Xmat[103X ) [32X operation[133X
  
  [33X[0;0YDoes the same as [2XDeterminantMatrix[102X ([14X24.4-4[114X), with the difference that it may
  destroy its argument. The matrix [3Xmat[103X must be mutable.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDeterminantMatrix([[1,2],[2,1]]);[127X[104X
    [4X[28X-3[128X[104X
    [4X[25Xgap>[125X [27Xmm:= [[1,2],[2,1]];;[127X[104X
    [4X[25Xgap>[125X [27XDeterminantMatrixDestructive( mm );[127X[104X
    [4X[28X-3[128X[104X
    [4X[25Xgap>[125X [27Xmm;[127X[104X
    [4X[28X[ [ 1, 2 ], [ 0, -3 ] ][128X[104X
  [4X[32X[104X
  
  [1X24.4-6 DeterminantMatrixDivFree[101X
  
  [33X[1;0Y[29X[2XDeterminantMatrixDivFree[102X( [3Xmat[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDeterminantMatDivFree[102X( [3Xmat[103X ) [32X operation[133X
  
  [33X[0;0Yreturn  the determinant of a square matrix [3Xmat[103X over an arbitrary commutative
  ring with one using the division free method of Mahajan and Vinay [MV97].[133X
  
  [1X24.4-7 IsEmptyMatrix[101X
  
  [33X[1;0Y[29X[2XIsEmptyMatrix[102X( [3XM[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10YA boolean[133X
  
  [33X[0;0YIs  [9Xtrue[109X  if  the  matrix object [3XM[103X either has zero columns or zero rows, and
  [9Xfalse[109X  otherwise.  In  other  words,  a  matrix object is empty if it has no
  entries.[133X
  
  [1X24.4-8 IsMonomialMatrix[101X
  
  [33X[1;0Y[29X[2XIsMonomialMatrix[102X( [3Xmat[103X ) [32X property[133X
  
  [33X[0;0YA  matrix  is  monomial  if  and only if it has exactly one nonzero entry in
  every row and every column.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsMonomialMatrix([[0,1],[1,0]]);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X24.4-9 IsDiagonalMatrix[101X
  
  [33X[1;0Y[29X[2XIsDiagonalMatrix[102X( [3Xmat[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsDiagonalMat[102X( [3Xmat[103X ) [32X property[133X
  
  [33X[0;0Yreturn  [9Xtrue[109X  if the matrix [3Xmat[103X has only zero entries off the main diagonal,
  and [9Xfalse[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsDiagonalMatrix( [ [ 1 ] ] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDiagonalMatrix( [ [ 1, 0, 0 ], [ 0, 1, 0 ] ] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsDiagonalMatrix( [ [ 0, 1 ], [ 1, 0 ] ] );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X24.4-10 IsUpperTriangularMatrix[101X
  
  [33X[1;0Y[29X[2XIsUpperTriangularMatrix[102X( [3Xmat[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsUpperTriangularMat[102X( [3Xmat[103X ) [32X property[133X
  
  [33X[0;0Yreturn [9Xtrue[109X if the matrix [3Xmat[103X has only zero entries below the main diagonal,
  and [9Xfalse[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsUpperTriangularMatrix( [ [ 1 ] ] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsUpperTriangularMatrix( [ [ 1, 2, 3 ], [ 0, 5, 6 ] ] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsUpperTriangularMatrix( [ [ 0, 1 ], [ 1, 0 ] ] );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X24.4-11 IsLowerTriangularMatrix[101X
  
  [33X[1;0Y[29X[2XIsLowerTriangularMatrix[102X( [3Xmat[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsLowerTriangularMat[102X( [3Xmat[103X ) [32X property[133X
  
  [33X[0;0Yreturn [9Xtrue[109X if the matrix [3Xmat[103X has only zero entries above the main diagonal,
  and [9Xfalse[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsLowerTriangularMatrix( [ [ 1 ] ] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsLowerTriangularMatrix( [ [ 1, 0, 0 ], [ 2, 3, 0 ] ] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsLowerTriangularMatrix( [ [ 0, 1 ], [ 1, 0 ] ] );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  
  [1X24.5 [33X[0;0YMatrix Constructions[133X[101X
  
  [1X24.5-1 IdentityMat[101X
  
  [33X[1;0Y[29X[2XIdentityMat[102X( [3Xm[103X[, [3XR[103X] ) [32X function[133X
  
  [33X[0;0Yreturns  a  (mutable)  [3Xm[103X[22X×[122X[3Xm[103X identity matrix over the ring given by [3XR[103X. Here, [3XR[103X
  can be either a ring, or an element of a ring. By default, an integer matrix
  is created.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIdentityMat(3);[127X[104X
    [4X[28X[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XIdentityMat(2,Integers mod 15);[127X[104X
    [4X[28X[ [ ZmodnZObj( 1, 15 ), ZmodnZObj( 0, 15 ) ], [128X[104X
    [4X[28X  [ ZmodnZObj( 0, 15 ), ZmodnZObj( 1, 15 ) ] ][128X[104X
    [4X[25Xgap>[125X [27XIdentityMat(2,Z(3));[127X[104X
    [4X[28X[ [ Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3)^0 ] ][128X[104X
  [4X[32X[104X
  
  [1X24.5-2 NullMat[101X
  
  [33X[1;0Y[29X[2XNullMat[102X( [3Xm[103X, [3Xn[103X[, [3XR[103X] ) [32X function[133X
  
  [33X[0;0Yreturns a (mutable) [3Xm[103X[22X×[122X[3Xn[103X null matrix over the ring given by by [3XR[103X. Here, [3XR[103X can
  be  either a ring, or an element of a ring. By default, an integer matrix is
  created.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNullMat(3,2);[127X[104X
    [4X[28X[ [ 0, 0 ], [ 0, 0 ], [ 0, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XNullMat(2,2,Integers mod 15);[127X[104X
    [4X[28X[ [ ZmodnZObj( 0, 15 ), ZmodnZObj( 0, 15 ) ], [128X[104X
    [4X[28X  [ ZmodnZObj( 0, 15 ), ZmodnZObj( 0, 15 ) ] ][128X[104X
    [4X[25Xgap>[125X [27XNullMat(3,2,Z(3));[127X[104X
    [4X[28X[ [ 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3) ] ][128X[104X
  [4X[32X[104X
  
  [1X24.5-3 EmptyMatrix[101X
  
  [33X[1;0Y[29X[2XEmptyMatrix[102X( [3Xchar[103X ) [32X function[133X
  
  [33X[0;0Yis an empty (ordinary) matrix in characteristic [3Xchar[103X that can be added to or
  multiplied  with empty lists (representing zero-dimensional row vectors). It
  also acts (via the operation [2X\^[102X ([14X31.12-1[114X)) on empty lists.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XEmptyMatrix(5);[127X[104X
    [4X[28XEmptyMatrix( 5 )[128X[104X
    [4X[25Xgap>[125X [27XAsList(last);[127X[104X
    [4X[28X[  ][128X[104X
  [4X[32X[104X
  
  [1X24.5-4 DiagonalMat[101X
  
  [33X[1;0Y[29X[2XDiagonalMat[102X( [3Xvector[103X ) [32X function[133X
  
  [33X[0;0Yreturns a diagonal matrix [3Xmat[103X with the diagonal entries given by [3Xvector[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDiagonalMat([1,2,3]);[127X[104X
    [4X[28X[ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 3 ] ][128X[104X
  [4X[32X[104X
  
  [1X24.5-5 PermutationMat[101X
  
  [33X[1;0Y[29X[2XPermutationMat[102X( [3Xperm[103X, [3Xdim[103X[, [3XF[103X] ) [32X function[133X
  
  [33X[0;0Yreturns  a  matrix  in  dimension  [3Xdim[103X  over  the field given by [3XF[103X (i.e. the
  smallest  field  containing the element [3XF[103X or [3XF[103X itself if it is a field) that
  represents  the permutation [3Xperm[103X acting by permuting the basis vectors as it
  permutes points.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPermutationMat((1,2,3),4,1);[127X[104X
    [4X[28X[ [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 1 ] ][128X[104X
  [4X[32X[104X
  
  [1X24.5-6 TransposedMatImmutable[101X
  
  [33X[1;0Y[29X[2XTransposedMatImmutable[102X( [3Xmat[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XTransposedMatAttr[102X( [3Xmat[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XTransposedMat[102X( [3Xmat[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XTransposedMatMutable[102X( [3Xmat[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XTransposedMatOp[102X( [3Xmat[103X ) [32X operation[133X
  
  [33X[0;0YThese  functions all return the transposed of the matrix object [3Xmat[103X, i.e., a
  matrix object [22Xtrans[122X such that [22Xtrans[i,k] = [3Xmat[103X[k,i][122X holds.[133X
  
  [33X[0;0YThey differ only w.r.t. the mutability of the result.[133X
  
  [33X[0;0Y[2XTransposedMat[102X  is  an  attribute  and  hence  returns  an  immutable result.
  [2XTransposedMatMutable[102X is guaranteed to return a new [13Xmutable[113X matrix.[133X
  
  [33X[0;0Y[2XTransposedMatImmutable[102X  and [2XTransposedMatAttr[102X are synonyms of [2XTransposedMat[102X,
  and  [2XTransposedMatOp[102X  is  a  synonym  of [2XTransposedMatMutable[102X, in analogy to
  operations such as [2XZero[102X ([14X31.10-3[114X).[133X
  
  [1X24.5-7 TransposedMatDestructive[101X
  
  [33X[1;0Y[29X[2XTransposedMatDestructive[102X( [3Xmat[103X ) [32X operation[133X
  
  [33X[0;0YIf  [3Xmat[103X is a mutable matrix, then the transposed is computed by swapping the
  entries  in  [3Xmat[103X.  In  this  way  [3Xmat[103X  gets  changed. In all other cases the
  transposed is computed by [2XTransposedMat[102X ([14X24.5-6[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XTransposedMat([[1,2,3],[4,5,6],[7,8,9]]);[127X[104X
    [4X[28X[ [ 1, 4, 7 ], [ 2, 5, 8 ], [ 3, 6, 9 ] ][128X[104X
    [4X[25Xgap>[125X [27Xmm:= [[1,2,3],[4,5,6],[7,8,9]];;[127X[104X
    [4X[25Xgap>[125X [27XTransposedMatDestructive( mm );[127X[104X
    [4X[28X[ [ 1, 4, 7 ], [ 2, 5, 8 ], [ 3, 6, 9 ] ][128X[104X
    [4X[25Xgap>[125X [27Xmm;[127X[104X
    [4X[28X[ [ 1, 4, 7 ], [ 2, 5, 8 ], [ 3, 6, 9 ] ][128X[104X
  [4X[32X[104X
  
  [1X24.5-8 KroneckerProduct[101X
  
  [33X[1;0Y[29X[2XKroneckerProduct[102X( [3Xmat1[103X, [3Xmat2[103X ) [32X operation[133X
  
  [33X[0;0YThe  Kronecker product of two matrices is the matrix obtained when replacing
  each entry [3Xa[103X of [3Xmat1[103X by the product [10X[3Xa[103X[10X*[3Xmat2[103X[10X[110X in one matrix.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XKroneckerProduct([[1,2]],[[5,7],[9,2]]);[127X[104X
    [4X[28X[ [ 5, 7, 10, 14 ], [ 9, 2, 18, 4 ] ][128X[104X
  [4X[32X[104X
  
  [1X24.5-9 ReflectionMat[101X
  
  [33X[1;0Y[29X[2XReflectionMat[102X( [3Xcoeffs[103X[, [3Xconj[103X][, [3Xroot[103X] ) [32X function[133X
  
  [33X[0;0YLet  [3Xcoeffs[103X  be  a  row  vector.  [2XReflectionMat[102X  returns  the  matrix of the
  reflection in this vector.[133X
  
  [33X[0;0YMore  precisely,  if  [3Xcoeffs[103X is the coefficients list of a vector [22Xv[122X w.r.t. a
  basis  [22XB[122X  (see [2XBasis[102X  ([14X61.5-2[114X))  then  the  returned  matrix  describes  the
  reflection  in  [22Xv[122X  w.r.t.  [22XB[122X  as  a map on a row space, with action from the
  right.[133X
  
  [33X[0;0YThe  optional  argument [3Xroot[103X 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 [2XE[102X ([14X18.1-1[114X)).[133X
  
  [33X[0;0Y[3Xconj[103X  is  a  function  of  one  argument that conjugates a ring element. The
  default is [2XComplexConjugate[102X ([14X18.5-2[114X).[133X
  
  [33X[0;0YThe matrix of the reflection in [22Xv[122X is defined as[133X
  
  
  [24X[33X[0;6YM = I_n + overline{v^tr} ⋅ (w-1) / ( v overline{v^tr} ) ⋅ v[133X[124X
  
  [33X[0;0Ywhere  [22Xw[122X  equals  [3Xroot[103X,  [22Xn[122X  is  the  length  of  the  coefficient  list, and
  [22Xoverline{vphantomx}[122X denotes the conjugation.[133X
  
  [33X[0;0YSo [22Xv[122X is mapped to [22Xw v[122X, with default [22X-v[122X, and any vector [22Xx[122X with the property [22Xx
  overline{v^tr} = 0[122X is fixed by the reflection.[133X
  
  [1X24.5-10 PrintArray[101X
  
  [33X[1;0Y[29X[2XPrintArray[102X( [3Xarray[103X ) [32X function[133X
  
  [33X[0;0Ypretty-prints the array [3Xarray[103X.[133X
  
  
  [1X24.6 [33X[0;0YRandom Matrices[133X[101X
  
  [1X24.6-1 RandomMat[101X
  
  [33X[1;0Y[29X[2XRandomMat[102X( [[3Xrs[103X, ][3Xm[103X, [3Xn[103X[, [3XR[103X] ) [32X function[133X
  
  [33X[0;0Y[2XRandomMat[102X returns a new mutable random matrix with [3Xm[103X rows and [3Xn[103X columns with
  elements taken from the ring [3XR[103X, which defaults to [2XIntegers[102X ([14X14[114X). Optionally,
  a random source [3Xrs[103X can be supplied.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRandomMat(2,3,GF(3));[127X[104X
    [4X[28X[ [ Z(3), Z(3), 0*Z(3) ], [ Z(3), Z(3)^0, Z(3) ] ][128X[104X
  [4X[32X[104X
  
  [1X24.6-2 RandomInvertibleMat[101X
  
  [33X[1;0Y[29X[2XRandomInvertibleMat[102X( [[3Xrs[103X, ][3Xm[103X[, [3XR[103X] ) [32X function[133X
  
  [33X[0;0Y[2XRandomInvertibleMat[102X  returns  a  new mutable invertible random matrix with [3Xm[103X
  rows  and  columns  with  elements  taken from the ring [3XR[103X, which defaults to
  [2XIntegers[102X ([14X14[114X). Optionally, a random source [3Xrs[103X can be supplied.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm := RandomInvertibleMat(4);[127X[104X
    [4X[28X[ [ -4, 1, 0, -1 ], [ -1, -1, 1, -1 ], [ 1, -2, -1, -2 ], [128X[104X
    [4X[28X  [ 0, -1, 2, -2 ] ][128X[104X
    [4X[25Xgap>[125X [27Xm^-1;[127X[104X
    [4X[28X[ [ -1/8, -11/24, 1/24, 1/4 ], [ 1/4, -13/12, -1/12, 1/2 ], [128X[104X
    [4X[28X  [ -1/8, 5/24, -7/24, 1/4 ], [ -1/4, 3/4, -1/4, -1/2 ] ][128X[104X
  [4X[32X[104X
  
  [1X24.6-3 RandomUnimodularMat[101X
  
  [33X[1;0Y[29X[2XRandomUnimodularMat[102X( [[3Xrs[103X, ][3Xm[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  new  random  mutable  [3Xm[103X[22X×[122X[3Xm[103X  matrix  with  integer entries that is
  invertible  over  the  integers.  Optionally,  a  random  source  [3Xrs[103X  can be
  supplied.  If  the  option  [3Xdomain[103X  is  given, random selection is made from
  [3Xdomain[103X, otherwise from [3XIntegers[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm := RandomUnimodularMat(3);[127X[104X
    [4X[28X[ [ -5, 1, 0 ], [ 12, -2, -1 ], [ -14, 3, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27Xm^-1;[127X[104X
    [4X[28X[ [ -3, 0, 1 ], [ -14, 0, 5 ], [ -8, -1, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27XRandomUnimodularMat(3:domain:=[-1000..1000]);[127X[104X
    [4X[28X[ [ 312330173, 15560030349, -125721926670 ],[128X[104X
    [4X[28X[ -307290, -15309014, 123693281 ],[128X[104X
    [4X[28X[ -684293792, -34090949551, 275448039848 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X24.7 [33X[0;0YMatrices Representing Linear Equations and the Gaussian Algorithm[133X[101X
  
  [1X24.7-1 RankMatrix[101X
  
  [33X[1;0Y[29X[2XRankMatrix[102X( [3Xmat[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XRankMat[102X( [3Xmat[103X ) [32X attribute[133X
  
  [33X[0;0YIf  [3Xmat[103X  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
  [2XRankMatrix[102X  returns  the  dimension  of  this free module. Otherwise [9Xfail[109X is
  returned.[133X
  
  [33X[0;0YNote  that [2XRankMatrix[102X may perform a Gaussian elimination. For large rational
  matrices this may take very long, because the entries may become very large.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat:=[[1,2,3],[4,5,6],[7,8,9]];;[127X[104X
    [4X[25Xgap>[125X [27XRankMatrix( mat );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [1X24.7-2 TriangulizedMat[101X
  
  [33X[1;0Y[29X[2XTriangulizedMat[102X( [3Xmat[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRREF[102X( [3Xmat[103X ) [32X operation[133X
  
  [33X[0;0YComputes  an  upper  triangular  form  of  the  matrix  [3Xmat[103X via the Gaussian
  Algorithm.  It  returns  a  mutable matrix in upper triangular form. This is
  sometimes  also called [21XHermite normal form[121X or [21XReduced Row Echelon Form[121X. [10XRREF[110X
  is a synonym for [10XTriangulizedMat[110X.[133X
  
  [1X24.7-3 TriangulizeMat[101X
  
  [33X[1;0Y[29X[2XTriangulizeMat[102X( [3Xmat[103X ) [32X operation[133X
  
  [33X[0;0YApplies  the  Gaussian  Algorithm  to the mutable matrix [3Xmat[103X and changes [3Xmat[103X
  such  that  it  is in upper triangular normal form (sometimes called [21XHermite
  normal form[121X or [21XReduced Row Echelon Form[121X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:=TransposedMatMutable(mat);[127X[104X
    [4X[28X[ [ 1, 4, 7 ], [ 2, 5, 8 ], [ 3, 6, 9 ] ][128X[104X
    [4X[25Xgap>[125X [27XTriangulizeMat(m);m;[127X[104X
    [4X[28X[ [ 1, 0, -1 ], [ 0, 1, 2 ], [ 0, 0, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27Xm:=TransposedMatMutable(mat);[127X[104X
    [4X[28X[ [ 1, 4, 7 ], [ 2, 5, 8 ], [ 3, 6, 9 ] ][128X[104X
    [4X[25Xgap>[125X [27XTriangulizedMat(m);m;[127X[104X
    [4X[28X[ [ 1, 0, -1 ], [ 0, 1, 2 ], [ 0, 0, 0 ] ][128X[104X
    [4X[28X[ [ 1, 4, 7 ], [ 2, 5, 8 ], [ 3, 6, 9 ] ][128X[104X
  [4X[32X[104X
  
  [1X24.7-4 NullspaceMat[101X
  
  [33X[1;0Y[29X[2XNullspaceMat[102X( [3Xmat[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XTriangulizedNullspaceMat[102X( [3Xmat[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns  a  list  of  row  vectors  that form a basis of the vector space of
  solutions to the equation [10X[3Xvec[103X[10X*[3Xmat[103X[10X=0[110X. The result is an immutable matrix. This
  basis is not guaranteed to be in any specific form.[133X
  
  [33X[0;0YThe  variant  [2XTriangulizedNullspaceMat[102X  returns  a basis of the nullspace in
  triangulized form as is often needed for algorithms.[133X
  
  [1X24.7-5 NullspaceMatDestructive[101X
  
  [33X[1;0Y[29X[2XNullspaceMatDestructive[102X( [3Xmat[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XTriangulizedNullspaceMatDestructive[102X( [3Xmat[103X ) [32X operation[133X
  
  [33X[0;0YThis  function  does  the same as [2XNullspaceMat[102X ([14X24.7-4[114X). However, the latter
  function  makes  a  copy  of [3Xmat[103X to avoid having to change it. This function
  does not do that; it returns the nullspace and may destroy [3Xmat[103X; this saves a
  lot of memory in case [3Xmat[103X is big. The matrix [3Xmat[103X must be mutable.[133X
  
  [33X[0;0YThe  variant  [2XTriangulizedNullspaceMatDestructive[102X  returns  a  basis  of the
  nullspace in triangulized form. It may destroy the matrix [3Xmat[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat:=[[1,2,3],[4,5,6],[7,8,9]];;[127X[104X
    [4X[25Xgap>[125X [27XNullspaceMat(mat);[127X[104X
    [4X[28X[ [ 1, -2, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xmm:=[[1,2,3],[4,5,6],[7,8,9]];;[127X[104X
    [4X[25Xgap>[125X [27XNullspaceMatDestructive( mm );[127X[104X
    [4X[28X[ [ 1, -2, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xmm;[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 0, -3, -6 ], [ 0, 0, 0 ] ][128X[104X
  [4X[32X[104X
  
  [1X24.7-6 SolutionMat[101X
  
  [33X[1;0Y[29X[2XSolutionMat[102X( [3Xmat[103X, [3Xvec[103X ) [32X operation[133X
  
  [33X[0;0Yreturns  a row vector [3Xx[103X that is a solution of the equation [10X[3Xx[103X[10X * [3Xmat[103X[10X = [3Xvec[103X[10X[110X. It
  returns [9Xfail[109X if no such vector exists.[133X
  
  [1X24.7-7 SolutionMatDestructive[101X
  
  [33X[1;0Y[29X[2XSolutionMatDestructive[102X( [3Xmat[103X, [3Xvec[103X ) [32X operation[133X
  
  [33X[0;0YDoes  the  same  as  [10XSolutionMat(  [3Xmat[103X[10X, [3Xvec[103X[10X )[110X except that it may destroy the
  matrix [3Xmat[103X and the vector [3Xvec[103X. The matrix [3Xmat[103X must be mutable.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat:=[[1,2,3],[4,5,6],[7,8,9]];;[127X[104X
    [4X[25Xgap>[125X [27XSolutionMat(mat,[3,5,7]);[127X[104X
    [4X[28X[ 5/3, 1/3, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xmm:= [[1,2,3],[4,5,6],[7,8,9]];;[127X[104X
    [4X[25Xgap>[125X [27Xv:= [3,5,7];;[127X[104X
    [4X[25Xgap>[125X [27XSolutionMatDestructive( mm, v );[127X[104X
    [4X[28X[ 5/3, 1/3, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xmm;[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 0, -3, -6 ], [ 0, 0, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27Xv;[127X[104X
    [4X[28X[ 0, 0, 0 ][128X[104X
  [4X[32X[104X
  
  [1X24.7-8 BaseFixedSpace[101X
  
  [33X[1;0Y[29X[2XBaseFixedSpace[102X( [3Xmats[103X ) [32X function[133X
  
  [33X[0;0Y[2XBaseFixedSpace[102X  returns a list of row vectors that form a base of the vector
  space  [22XV[122X  such  that  [22Xv  M = v[122X for all [22Xv[122X in [22XV[122X and all matrices [22XM[122X in the list
  [3Xmats[103X.  (This  is  the  common  eigenspace  of  all  matrices in [3Xmats[103X for the
  eigenvalue 1.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBaseFixedSpace([[[1,2],[0,1]]]);[127X[104X
    [4X[28X[ [ 0, 1 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X24.8 [33X[0;0YEigenvectors and eigenvalues[133X[101X
  
  [1X24.8-1 GeneralisedEigenvalues[101X
  
  [33X[1;0Y[29X[2XGeneralisedEigenvalues[102X( [3XF[103X, [3XA[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XGeneralizedEigenvalues[102X( [3XF[103X, [3XA[103X ) [32X operation[133X
  
  [33X[0;0YThe generalised eigenvalues of the matrix [3XA[103X over the field [3XF[103X.[133X
  
  [1X24.8-2 GeneralisedEigenspaces[101X
  
  [33X[1;0Y[29X[2XGeneralisedEigenspaces[102X( [3XF[103X, [3XA[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XGeneralizedEigenspaces[102X( [3XF[103X, [3XA[103X ) [32X operation[133X
  
  [33X[0;0YThe generalised eigenspaces of the matrix [3XA[103X over the field [3XF[103X.[133X
  
  [1X24.8-3 Eigenvalues[101X
  
  [33X[1;0Y[29X[2XEigenvalues[102X( [3XF[103X, [3XA[103X ) [32X operation[133X
  
  [33X[0;0YThe eigenvalues of the matrix [3XA[103X over the field [3XF[103X.[133X
  
  [1X24.8-4 Eigenspaces[101X
  
  [33X[1;0Y[29X[2XEigenspaces[102X( [3XF[103X, [3XA[103X ) [32X operation[133X
  
  [33X[0;0YThe eigenspaces of the matrix [3XA[103X over the field [3XF[103X.[133X
  
  [1X24.8-5 Eigenvectors[101X
  
  [33X[1;0Y[29X[2XEigenvectors[102X( [3XF[103X, [3XA[103X ) [32X operation[133X
  
  [33X[0;0YThe eigenvectors of the matrix [3XA[103X over the field [3XF[103X.[133X
  
  
  [1X24.9 [33X[0;0YElementary Divisors[133X[101X
  
  [33X[0;0YSee also chapter [14X25[114X.[133X
  
  [1X24.9-1 ElementaryDivisorsMat[101X
  
  [33X[1;0Y[29X[2XElementaryDivisorsMat[102X( [[3Xring[103X, ][3Xmat[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XElementaryDivisorsMatDestructive[102X( [3Xring[103X, [3Xmat[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  list  of  the  elementary divisors, i.e., the unique [22Xd[122X with [22Xd[i][122X
  divides  [22Xd[i+1][122X and [3Xmat[103X is equivalent to a diagonal matrix with the elements
  [22Xd[i][122X  on  the diagonal. The operations are performed over the euclidean ring
  [3Xring[103X,  which  must  contain all matrix entries. For compatibility reasons it
  can  be  omitted  and  defaults  to  the  [2XDefaultRing[102X ([14X56.1-3[114X) of the matrix
  entries.[133X
  
  [33X[0;0YThe  function  [2XElementaryDivisorsMatDestructive[102X produces the same result but
  in the process may destroy the contents of [3Xmat[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat:=[[1,2,3],[4,5,6],[7,8,9]];;[127X[104X
    [4X[25Xgap>[125X [27XElementaryDivisorsMat(mat);[127X[104X
    [4X[28X[ 1, 3, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xx:=Indeterminate(Rationals,"x");;[127X[104X
    [4X[25Xgap>[125X [27Xmat:=mat*One(x)-x*mat^0;       [127X[104X
    [4X[28X[ [ -x+1, 2, 3 ], [ 4, -x+5, 6 ], [ 7, 8, -x+9 ] ][128X[104X
    [4X[25Xgap>[125X [27XElementaryDivisorsMat(PolynomialRing(Rationals,1),mat);[127X[104X
    [4X[28X[ 1, 1, x^3-15*x^2-18*x ][128X[104X
    [4X[25Xgap>[125X [27Xmat:=KroneckerProduct(CompanionMat((x-1)^2),[127X[104X
    [4X[25X>[125X [27X                         CompanionMat((x^3-1)*(x-1)));;[127X[104X
    [4X[25Xgap>[125X [27Xmat:=mat*One(x)-x*mat^0;[127X[104X
    [4X[28X[ [ -x, 0, 0, 0, 0, 0, 0, 1 ], [ 0, -x, 0, 0, -1, 0, 0, -1 ], [128X[104X
    [4X[28X  [ 0, 0, -x, 0, 0, -1, 0, 0 ], [ 0, 0, 0, -x, 0, 0, -1, -1 ], [128X[104X
    [4X[28X  [ 0, 0, 0, -1, -x, 0, 0, -2 ], [ 1, 0, 0, 1, 2, -x, 0, 2 ], [128X[104X
    [4X[28X  [ 0, 1, 0, 0, 0, 2, -x, 0 ], [ 0, 0, 1, 1, 0, 0, 2, -x+2 ] ][128X[104X
    [4X[25Xgap>[125X [27XElementaryDivisorsMat(PolynomialRing(Rationals,1),mat);[127X[104X
    [4X[28X[ 1, 1, 1, 1, 1, 1, x-1, x^7-x^6-2*x^4+2*x^3+x-1 ][128X[104X
  [4X[32X[104X
  
  [1X24.9-2 ElementaryDivisorsTransformationsMat[101X
  
  [33X[1;0Y[29X[2XElementaryDivisorsTransformationsMat[102X( [[3Xring[103X, ][3Xmat[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XElementaryDivisorsTransformationsMatDestructive[102X( [3Xring[103X, [3Xmat[103X ) [32X function[133X
  
  [33X[0;0Y[10XElementaryDivisorsTransformations[110X,   in   addition  to  the  tasks  done  by
  [10XElementaryDivisorsMat[110X,  also  calculates transforming matrices. It returns a
  record  with  components  [10Xnormal[110X  (a  matrix  [22XS[122X), [10Xrowtrans[110X (a matrix [22XP[122X), and
  [10Xcoltrans[110X (a matrix [22XQ[122X) such that [22XP A Q = S[122X. The operations are performed over
  the  euclidean  ring  [3Xring[103X,  which  must  contain  all  matrix  entries. For
  compatibility  reasons  it  can  be  omitted and defaults to the [2XDefaultRing[102X
  ([14X56.1-3[114X) of the matrix entries.[133X
  
  [33X[0;0YThe  function  [2XElementaryDivisorsTransformationsMatDestructive[102X  produces the
  same result but in the process destroys the contents of [3Xmat[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat:=KroneckerProduct(CompanionMat((x-1)^2),CompanionMat((x^3-1)*(x-1)));;[127X[104X
    [4X[25Xgap>[125X [27Xmat:=mat*One(x)-x*mat^0;[127X[104X
    [4X[28X[ [ -x, 0, 0, 0, 0, 0, 0, 1 ], [ 0, -x, 0, 0, -1, 0, 0, -1 ], [128X[104X
    [4X[28X  [ 0, 0, -x, 0, 0, -1, 0, 0 ], [ 0, 0, 0, -x, 0, 0, -1, -1 ], [128X[104X
    [4X[28X  [ 0, 0, 0, -1, -x, 0, 0, -2 ], [ 1, 0, 0, 1, 2, -x, 0, 2 ], [128X[104X
    [4X[28X  [ 0, 1, 0, 0, 0, 2, -x, 0 ], [ 0, 0, 1, 1, 0, 0, 2, -x+2 ] ][128X[104X
    [4X[25Xgap>[125X [27Xt:=ElementaryDivisorsTransformationsMat(PolynomialRing(Rationals,1),mat);[127X[104X
    [4X[28Xrec( coltrans := [ [ 0, 0, 0, 0, 0, 0, 1/6*x^2-7/9*x-1/18, -3*x^3-x^2-x-1 ], [128X[104X
    [4X[28X      [ 0, 0, 0, 0, 0, 0, -1/6*x^2+x-1, 3*x^3-3*x^2 ], [128X[104X
    [4X[28X      [ 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 [128X[104X
    [4X[28X         ], [ 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 ],[128X[104X
    [4X[28X      [ 0, -1, 0, 0, 0, 0, -2/9*x^2+19/18*x, 4*x^3+x^2+x ], [128X[104X
    [4X[28X      [ 0, 0, -1, 0, 0, -x, 1/18*x^5-1/3*x^4+1/3*x^3+1/9*x^2, [128X[104X
    [4X[28X          -x^6+x^5-2*x^3+2*x^2 ], [128X[104X
    [4X[28X      [ 0, 0, 0, -1, x, 0, 1/9*x^4-2/3*x^3+2/3*x^2+1/18*x, [128X[104X
    [4X[28X          -2*x^5+2*x^4-x^2+x ], [128X[104X
    [4X[28X      [ 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 ] ], [128X[104X
    [4X[28X  normal := [ [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X      [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [128X[104X
    [4X[28X      [ 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], [128X[104X
    [4X[28X      [ 0, 0, 0, 0, 0, 0, x-1, 0 ], [128X[104X
    [4X[28X      [ 0, 0, 0, 0, 0, 0, 0, x^7-x^6-2*x^4+2*x^3+x-1 ] ], [128X[104X
    [4X[28X  rowtrans := [ [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 1, 1, 0, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X      [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 1, 0, 0, 0, 0 ], [128X[104X
    [4X[28X      [ -x+2, -x, 0, 0, 1, 0, 0, 0 ], [128X[104X
    [4X[28X      [ 2*x^2-4*x+2, 2*x^2-x, 0, 2, -2*x+1, 0, 0, 1 ], [128X[104X
    [4X[28X      [ 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 ], [128X[104X
    [4X[28X      [ 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, [128X[104X
    [4X[28X          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+\[128X[104X
    [4X[28X2, 1/9*x^5-5/9*x^4+1/9*x^3-1/9*x^2+14/9*x-1/9, [128X[104X
    [4X[28X          1/6*x^6-5/6*x^5+1/6*x^4-1/6*x^3+11/6*x^2-1/6*x, [128X[104X
    [4X[28X          -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, [128X[104X
    [4X[28X          1, 1/18*x^5-5/18*x^4+1/18*x^3-1/18*x^2+23/18*x-1/18, [128X[104X
    [4X[28X          1/9*x^6-5/9*x^5+1/9*x^4-1/9*x^3+14/9*x^2-1/9*x ] ] )[128X[104X
    [4X[25Xgap>[125X [27Xt.rowtrans*mat*t.coltrans;[127X[104X
    [4X[28X[ [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X  [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [128X[104X
    [4X[28X  [ 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], [128X[104X
    [4X[28X  [ 0, 0, 0, 0, 0, 0, x-1, 0 ], [128X[104X
    [4X[28X  [ 0, 0, 0, 0, 0, 0, 0, x^7-x^6-2*x^4+2*x^3+x-1 ] ][128X[104X
  [4X[32X[104X
  
  [1X24.9-3 DiagonalizeMat[101X
  
  [33X[1;0Y[29X[2XDiagonalizeMat[102X( [3Xring[103X, [3Xmat[103X ) [32X operation[133X
  
  [33X[0;0Ybrings  the  mutable  matrix  [3Xmat[103X,  considered  as  a matrix over [3Xring[103X, into
  diagonal form by elementary row and column operations.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:=[[1,2],[2,1]];;[127X[104X
    [4X[25Xgap>[125X [27XDiagonalizeMat(Integers,m);m;[127X[104X
    [4X[28X[ [ 1, 0 ], [ 0, 3 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X24.10 [33X[0;0YEchelonized Matrices[133X[101X
  
  [1X24.10-1 SemiEchelonMat[101X
  
  [33X[1;0Y[29X[2XSemiEchelonMat[102X( [3Xmat[103X ) [32X attribute[133X
  
  [33X[0;0YA matrix over a field [22XF[122X is in semi-echelon form if the first nonzero element
  in  each row is the identity of [22XF[122X, and all values exactly below these pivots
  are the zero of [22XF[122X.[133X
  
  [33X[0;0Y[2XSemiEchelonMat[102X   returns   a   record  that  contains  information  about  a
  semi-echelonized form of the matrix [3Xmat[103X.[133X
  
  [33X[0;0YThe components of this record are[133X
  
  [8X[10Xvectors[110X[8X[108X
        [33X[0;6Ylist of row vectors, each with pivot element the identity of [22XF[122X,[133X
  
  [8X[10Xheads[110X[8X[108X
        [33X[0;6Ylist  that  contains  at position [3Xi[103X, if nonzero, the number of the row
        for that the pivot element is in column [3Xi[103X.[133X
  
  [1X24.10-2 SemiEchelonMatDestructive[101X
  
  [33X[1;0Y[29X[2XSemiEchelonMatDestructive[102X( [3Xmat[103X ) [32X operation[133X
  
  [33X[0;0YThis  does  the  same  as  [10XSemiEchelonMat(  [3Xmat[103X[10X  )[110X,  except that it may (and
  probably will) destroy the matrix [3Xmat[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmm:=[[1,2,3],[4,5,6],[7,8,9]];;[127X[104X
    [4X[25Xgap>[125X [27XSemiEchelonMatDestructive( mm );[127X[104X
    [4X[28Xrec( heads := [ 1, 2, 0 ], vectors := [ [ 1, 2, 3 ], [ 0, 1, 2 ] ] )[128X[104X
    [4X[25Xgap>[125X [27Xmm;[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 0, 1, 2 ], [ 0, 0, 0 ] ][128X[104X
  [4X[32X[104X
  
  [1X24.10-3 SemiEchelonMatTransformation[101X
  
  [33X[1;0Y[29X[2XSemiEchelonMatTransformation[102X( [3Xmat[103X ) [32X attribute[133X
  
  [33X[0;0Ydoes the same as [2XSemiEchelonMat[102X ([14X24.10-1[114X) but additionally stores the linear
  transformation  [22XT[122X  performed on the matrix. The additional components of the
  result are[133X
  
  [8X[10Xcoeffs[110X[8X[108X
        [33X[0;6Ya  list of coefficients vectors of the [10Xvectors[110X component, with respect
        to the rows of [3Xmat[103X, that is, [10Xcoeffs * mat[110X is the [10Xvectors[110X component.[133X
  
  [8X[10Xrelations[110X[8X[108X
        [33X[0;6Ya list of basis vectors for the (left) null space of [3Xmat[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSemiEchelonMatTransformation([[1,2,3],[0,0,1]]);[127X[104X
    [4X[28Xrec( coeffs := [ [ 1, 0 ], [ 0, 1 ] ], heads := [ 1, 0, 2 ], [128X[104X
    [4X[28X  relations := [  ], vectors := [ [ 1, 2, 3 ], [ 0, 0, 1 ] ] )[128X[104X
  [4X[32X[104X
  
  [1X24.10-4 SemiEchelonMats[101X
  
  [33X[1;0Y[29X[2XSemiEchelonMats[102X( [3Xmats[103X ) [32X operation[133X
  
  [33X[0;0YA list of matrices over a field [22XF[122X 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 [2XSemiEchelonMat[102X ([14X24.10-1[114X)).[133X
  
  [33X[0;0Y[2XSemiEchelonMats[102X   returns   a  record  that  contains  information  about  a
  semi-echelonized form of the list [3Xmats[103X of matrices.[133X
  
  [33X[0;0YThe components of this record are[133X
  
  [8X[10Xvectors[110X[8X[108X
        [33X[0;6Ylist of matrices, each with pivot element the identity of [22XF[122X,[133X
  
  [8X[10Xheads[110X[8X[108X
        [33X[0;6Ymatrix  that contains at position [[3Xi[103X,[3Xj[103X], if nonzero, the number of the
        matrix that has the pivot element in this position[133X
  
  [1X24.10-5 SemiEchelonMatsDestructive[101X
  
  [33X[1;0Y[29X[2XSemiEchelonMatsDestructive[102X( [3Xmats[103X ) [32X operation[133X
  
  [33X[0;0YDoes  the  same as [2XSemiEchelonMats[102X ([14X24.10-4[114X), except that it may destroy its
  argument.  Therefore  the  argument  must  be  a  list  of matrices that are
  mutable.[133X
  
  
  [1X24.11 [33X[0;0YMatrices as Basis of a Row Space[133X[101X
  
  [33X[0;0YSee also chapter [14X25[114X[133X
  
  [1X24.11-1 BaseMat[101X
  
  [33X[1;0Y[29X[2XBaseMat[102X( [3Xmat[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns  a  basis for the row space generated by the rows of [3Xmat[103X in the form
  of an immutable matrix.[133X
  
  [1X24.11-2 BaseMatDestructive[101X
  
  [33X[1;0Y[29X[2XBaseMatDestructive[102X( [3Xmat[103X ) [32X operation[133X
  
  [33X[0;0YDoes  the same as [2XBaseMat[102X ([14X24.11-1[114X), with the difference that it may destroy
  the matrix [3Xmat[103X. The matrix [3Xmat[103X must be mutable.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat:=[[1,2,3],[4,5,6],[7,8,9]];;[127X[104X
    [4X[25Xgap>[125X [27XBaseMat(mat);[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 0, 1, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27Xmm:= [[1,2,3],[4,5,6],[5,7,9]];;[127X[104X
    [4X[25Xgap>[125X [27XBaseMatDestructive( mm );[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 0, 1, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27Xmm;[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 0, 1, 2 ], [ 0, 0, 0 ] ][128X[104X
  [4X[32X[104X
  
  [1X24.11-3 BaseOrthogonalSpaceMat[101X
  
  [33X[1;0Y[29X[2XBaseOrthogonalSpaceMat[102X( [3Xmat[103X ) [32X attribute[133X
  
  [33X[0;0YLet  [22XV[122X  be  the  row space generated by the rows of [3Xmat[103X (over any field that
  contains  all entries of [3Xmat[103X). [10XBaseOrthogonalSpaceMat( [3Xmat[103X[10X )[110X computes a base
  of the orthogonal space of [22XV[122X.[133X
  
  [33X[0;0YThe rows of [3Xmat[103X need not be linearly independent.[133X
  
  [1X24.11-4 SumIntersectionMat[101X
  
  [33X[1;0Y[29X[2XSumIntersectionMat[102X( [3XM1[103X, [3XM2[103X ) [32X operation[133X
  
  [33X[0;0Yperforms  Zassenhaus'  algorithm  to  compute  bases  for  the  sum  and the
  intersection of spaces generated by the rows of the matrices [3XM1[103X, [3XM2[103X.[133X
  
  [33X[0;0Yreturns  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 [14X24.10[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSumIntersectionMat(mat,[[2,7,6],[5,9,4]]);[127X[104X
    [4X[28X[ [ [ 1, 2, 3 ], [ 0, 1, 2 ], [ 0, 0, 1 ] ], [ [ 1, -3/4, -5/2 ] ] ][128X[104X
  [4X[32X[104X
  
  [1X24.11-5 BaseSteinitzVectors[101X
  
  [33X[1;0Y[29X[2XBaseSteinitzVectors[102X( [3Xbas[103X, [3Xmat[103X ) [32X function[133X
  
  [33X[0;0Yfind vectors extending mat to a basis spanning the span of [3Xbas[103X. Both [3Xbas[103X and
  [3Xmat[103X  must  be  matrices  of  full  (row)  rank. It returns a record with the
  following components:[133X
  
  [8X[10Xsubspace[110X[8X[108X
        [33X[0;6Ys  a  basis  of the space spanned by [3Xmat[103X in upper triangular form with
        leading ones at all echelon steps and zeroes above these ones.[133X
  
  [8X[10Xfactorspace[110X[8X[108X
        [33X[0;6Yis a list of extending vectors in upper triangular form.[133X
  
  [8X[10Xfactorzero[110X[8X[108X
        [33X[0;6Yis a zero vector.[133X
  
  [8X[10Xheads[110X[8X[108X
        [33X[0;6Yis  a  list  of integers which can be used to decompose vectors in the
        basis  vectors.  The  [3Xi[103Xth  entry  indicating  the vector that gives an
        echelon  step  at  position  [3Xi[103X. 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 [10Xsubspace[110X and [10Xfactorspace[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBaseSteinitzVectors(IdentityMat(3,1),[[11,13,15]]);[127X[104X
    [4X[28Xrec( factorspace := [ [ 0, 1, 15/13 ], [ 0, 0, 1 ] ], [128X[104X
    [4X[28X  factorzero := [ 0, 0, 0 ], heads := [ -1, 1, 2 ], [128X[104X
    [4X[28X  subspace := [ [ 1, 13/11, 15/11 ] ] )[128X[104X
  [4X[32X[104X
  
  
  [1X24.12 [33X[0;0YTriangular Matrices[133X[101X
  
  [1X24.12-1 DiagonalOfMatrix[101X
  
  [33X[1;0Y[29X[2XDiagonalOfMatrix[102X( [3Xmat[103X ) [32X function[133X
  [33X[1;0Y[29X[2XDiagonalOfMat[102X( [3Xmat[103X ) [32X function[133X
  
  [33X[0;0Yreturn  the  diagonal of the matrix [3Xmat[103X. If [3Xmat[103X is not a square matrix, then
  the  result has the same length as the rows of [3Xmat[103X, and is padded with zeros
  if [3Xmat[103X has fewer rows than columns.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDiagonalOfMatrix( [ [ 1, 2, 3 ], [ 4, 5, 6 ] ] );[127X[104X
    [4X[28X[ 1, 5, 0 ][128X[104X
  [4X[32X[104X
  
  [1X24.12-2 UpperSubdiagonal[101X
  
  [33X[1;0Y[29X[2XUpperSubdiagonal[102X( [3Xmat[103X, [3Xpos[103X ) [32X operation[133X
  
  [33X[0;0Yreturns a mutable list containing the entries of the [3Xpos[103Xth upper subdiagonal
  of the matrix [3Xmat[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XUpperSubdiagonal( [ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ], 1 );[127X[104X
    [4X[28X[ 2, 6 ][128X[104X
    [4X[25Xgap>[125X [27XUpperSubdiagonal( [ [ 1, 2 ], [ 3, 4 ], [ 5, 6 ] ], 1 );[127X[104X
    [4X[28X[ 2 ][128X[104X
    [4X[25Xgap>[125X [27XUpperSubdiagonal( [ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8 ] ], 1 );[127X[104X
    [4X[28X[ 2, 7 ][128X[104X
  [4X[32X[104X
  
  [1X24.12-3 DepthOfUpperTriangularMatrix[101X
  
  [33X[1;0Y[29X[2XDepthOfUpperTriangularMatrix[102X( [3Xmat[103X ) [32X attribute[133X
  
  [33X[0;0YIf [3Xmat[103X is an upper triangular matrix this attribute returns the index of the
  first nonzero diagonal.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDepthOfUpperTriangularMatrix([[0,1,2],[0,0,1],[0,0,0]]);[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XDepthOfUpperTriangularMatrix([[0,0,2],[0,0,0],[0,0,0]]);[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  
  [1X24.13 [33X[0;0YMatrices as Linear Mappings[133X[101X
  
  [1X24.13-1 CharacteristicPolynomial[101X
  
  [33X[1;0Y[29X[2XCharacteristicPolynomial[102X( [[3XF[103X, [3XE[103X, ][3Xmat[103X[, [3Xind[103X] ) [32X attribute[133X
  
  [33X[0;0YFor a square matrix [3Xmat[103X, [2XCharacteristicPolynomial[102X returns the [13Xcharacteristic
  polynomial[113X   of   [3Xmat[103X,  that  is,  the  [2XStandardAssociate[102X  ([14X56.5-5[114X)  of  the
  determinant  of the matrix [22X[3Xmat[103X - X ⋅ I[122X, where [22XX[122X is an indeterminate and [22XI[122X is
  the appropriate identity matrix.[133X
  
  [33X[0;0YIf  fields  [3XF[103X  and [3XE[103X are given, then [3XF[103X must be a subfield of [3XE[103X, and [3Xmat[103X must
  have  entries in [3XE[103X. Then [2XCharacteristicPolynomial[102X returns the characteristic
  polynomial of the [3XF[103X-linear mapping induced by [3Xmat[103X on the underlying [3XE[103X-vector
  space  of [3Xmat[103X. In this case, the characteristic polynomial is computed using
  [2XBlownUpMat[102X ([14X24.13-4[114X) for the field extension of [22XE/F[122X generated by the default
  field.  Thus,  if  [22XF  =  E[122X,  the  result is the same as for the one argument
  version.[133X
  
  [33X[0;0YThe  returned  polynomials are expressed in the indeterminate number [3Xind[103X. If
  [3Xind[103X is not given, it defaults to [22X1[122X.[133X
  
  [33X[0;0Y[10XCharacteristicPolynomial([3XF[103X[10X,  [3XE[103X[10X, [3Xmat[103X[10X)[110X is a multiple of the minimal polynomial
  [10XMinimalPolynomial([3XF[103X[10X, [3Xmat[103X[10X)[110X (see [2XMinimalPolynomial[102X ([14X66.8-1[114X)).[133X
  
  [33X[0;0YNote that, up to [5XGAP[105X version 4.4.6, [2XCharacteristicPolynomial[102X only allowed to
  specify  one  field (corresponding to [3XF[103X) 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 [3XF[103X contains the default field of the matrix,
  see [2XDefaultFieldOfMatrix[102X ([14X24.4-2[114X), but this feature will disappear in future
  versions of [5XGAP[105X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCharacteristicPolynomial( [ [ 1, 1 ], [ 0, 1 ] ] );[127X[104X
    [4X[28Xx^2-2*x+1[128X[104X
    [4X[25Xgap>[125X [27Xmat := [[0,1],[E(4)-1,E(4)]];;[127X[104X
    [4X[25Xgap>[125X [27XCharacteristicPolynomial( mat );[127X[104X
    [4X[28Xx^2+(-E(4))*x+(1-E(4))[128X[104X
    [4X[25Xgap>[125X [27XCharacteristicPolynomial( Rationals, CF(4), mat );[127X[104X
    [4X[28Xx^4+3*x^2+2*x+2[128X[104X
    [4X[25Xgap>[125X [27Xmat:= [ [ E(4), 1 ], [ 0, -E(4) ] ];;[127X[104X
    [4X[25Xgap>[125X [27XCharacteristicPolynomial( mat );[127X[104X
    [4X[28Xx^2+1[128X[104X
    [4X[25Xgap>[125X [27XCharacteristicPolynomial( Rationals, CF(4), mat );[127X[104X
    [4X[28Xx^4+2*x^2+1[128X[104X
  [4X[32X[104X
  
  [1X24.13-2 RationalCanonicalFormTransform[101X
  
  [33X[1;0Y[29X[2XRationalCanonicalFormTransform[102X( [3Xmat[103X ) [32X function[133X
  
  [33X[0;0YFor  a  matrix  [10XA[110X,  return a matrix [10XP[110X such that [22XA^P[122X is in rational canonical
  form  (also  called  Frobenius normal form). The algorithm used is the basic
  textbook version and thus not of optimal complexity.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xaa:=[[0,-8,12,40,-36,4,0,59,15,-9],[-2,-2,-2,6,-11,1,-1,10,1,0],[127X[104X
    [4X[25X>[125X [27X[1,5,0,-6,12,-2,0,-12,-4,2],[0,0,0,2,0,0,0,7,0,0],[127X[104X
    [4X[25X>[125X [27X[0,2,-3,-7,8,-1,0,-7,-3,2],[-5,-4,-6,18,-30,2,-2,35,5,-1],[127X[104X
    [4X[25X>[125X [27X[-1,-6,6,20,-28,3,0,24,10,-6],[0,0,0,-1,0,0,0,-3,0,0],[127X[104X
    [4X[25X>[125X [27X[0,0,-1,-2,-2,0,-1,-7,0,0],[0,-8,9,21,-36,4,-2,12,12,-8]];;[127X[104X
    [4X[25Xgap>[125X [27Xt:=RationalCanonicalFormTransform(aa);;[127X[104X
    [4X[25Xgap>[125X [27Xaa^t;[127X[104X
    [4X[28X[ [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X  [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X  [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 1 ],[128X[104X
    [4X[28X  [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],[128X[104X
    [4X[28X  [ 0, 0, 0, 0, 0, 0, 0, 1, 0, -1 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1, -1 ] ][128X[104X
  [4X[32X[104X
  
  [1X24.13-3 JordanDecomposition[101X
  
  [33X[1;0Y[29X[2XJordanDecomposition[102X( [3Xmat[103X ) [32X attribute[133X
  
  [33X[0;0Y[10XJordanDecomposition( [3Xmat [103X[10X )[110X returns a list [10X[S,N][110X such that [10XS[110X is a semisimple
  matrix and [10XN[110X is nilpotent. Furthermore, [10XS[110X and [10XN[110X commute and [10X[3Xmat[103X[10X=S+N[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat:=[[1,2,3],[4,5,6],[7,8,9]];;[127X[104X
    [4X[25Xgap>[125X [27XJordanDecomposition(mat);[127X[104X
    [4X[28X[ [ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ], [128X[104X
    [4X[28X  [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ][128X[104X
  [4X[32X[104X
  
  [1X24.13-4 BlownUpMat[101X
  
  [33X[1;0Y[29X[2XBlownUpMat[102X( [3XB[103X, [3Xmat[103X ) [32X function[133X
  
  [33X[0;0YLet  [3XB[103X be a basis of a field extension [22XF / K[122X, and [3Xmat[103X a matrix whose entries
  are all in [22XF[122X. (This is not checked.) [2XBlownUpMat[102X returns a matrix over [22XK[122X that
  is  obtained  by  replacing  each entry of [3Xmat[103X by its regular representation
  w.r.t. [3XB[103X.[133X
  
  [33X[0;0YMore  precisely,  regard [3Xmat[103X as the matrix of a linear transformation on the
  row  space [22XF^n[122X w.r.t. the [22XF[122X-basis with vectors [22X(v_1, ldots, v_n)[122X and suppose
  that  the basis [3XB[103X consists of the vectors [22X(b_1, ..., b_m)[122X; then the returned
  matrix  is  the  matrix  of  the linear transformation on the row space [22XK^mn[122X
  w.r.t. the [22XK[122X-basis whose vectors are [22X(b_1 v_1, ... b_m v_1, ..., b_m v_n)[122X.[133X
  
  [33X[0;0YNote  that  the linear transformations act on [13Xrow[113X vectors, i.e., each row of
  the matrix is a concatenation of vectors of [3XB[103X-coefficients.[133X
  
  [1X24.13-5 BlownUpVector[101X
  
  [33X[1;0Y[29X[2XBlownUpVector[102X( [3XB[103X, [3Xvector[103X ) [32X function[133X
  
  [33X[0;0YLet  [3XB[103X  be a basis of a field extension [22XF / K[122X, and [3Xvector[103X a row vector whose
  entries  are  all  in  [22XF[122X.  [2XBlownUpVector[102X returns a row vector over [22XK[122X that is
  obtained by replacing each entry of [3Xvector[103X by its coefficients w.r.t. [3XB[103X.[133X
  
  [33X[0;0YSo  [2XBlownUpVector[102X  and [2XBlownUpMat[102X ([14X24.13-4[114X) are compatible in the sense that
  for  a  matrix  [3Xmat[103X  over  [22XF[122X,  [10XBlownUpVector(  [3XB[103X[10X, [3Xmat[103X[10X * [3Xvector[103X[10X )[110X is equal to
  [10XBlownUpMat( [3XB[103X[10X, [3Xmat[103X[10X ) * BlownUpVector( [3XB[103X[10X, [3Xvector[103X[10X )[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XB:= Basis( CF(4), [ 1, E(4) ] );;[127X[104X
    [4X[25Xgap>[125X [27Xmat:= [ [ 1, E(4) ], [ 0, 1 ] ];;  vec:= [ 1, E(4) ];;[127X[104X
    [4X[25Xgap>[125X [27Xbmat:= BlownUpMat( B, mat );;  bvec:= BlownUpVector( B, vec );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( bmat );  bvec;[127X[104X
    [4X[28X[ [   1,   0,   0,   1 ],[128X[104X
    [4X[28X  [   0,   1,  -1,   0 ],[128X[104X
    [4X[28X  [   0,   0,   1,   0 ],[128X[104X
    [4X[28X  [   0,   0,   0,   1 ] ][128X[104X
    [4X[28X[ 1, 0, 0, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xbvec * bmat = BlownUpVector( B, vec * mat );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X24.13-6 CompanionMatrix[101X
  
  [33X[1;0Y[29X[2XCompanionMatrix[102X( [3Xpoly[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XCompanionMat[102X( [3Xpoly[103X ) [32X operation[133X
  
  [33X[0;0Ycompute  a  companion matrix of the polynomial [3Xpoly[103X. This matrix has [3Xpoly[103X as
  its minimal polynomial.[133X
  
  
  [1X24.14 [33X[0;0YMatrices over Finite Fields[133X[101X
  
  [33X[0;0YJust   as   for   row  vectors,  (see  section  [14X23.3[114X),  [5XGAP[105X  has  a  special
  representation for matrices over small finite fields.[133X
  
  [33X[0;0YTo  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 [13Xthe
  same[113X finite field.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xv := Z(2)*[1,0,0,1,1];[127X[104X
    [4X[28X[ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ][128X[104X
    [4X[25Xgap>[125X [27XConvertToVectorRep(v,2);[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xv;[127X[104X
    [4X[28X<a GF2 vector of length 5>[128X[104X
    [4X[25Xgap>[125X [27Xm := [v];; ConvertToMatrixRep(m,GF(2));; m;[127X[104X
    [4X[28X<a 1x5 matrix over GF2>[128X[104X
    [4X[25Xgap>[125X [27Xm := [v,v];; ConvertToMatrixRep(m,GF(2));; m;[127X[104X
    [4X[28X<a 2x5 matrix over GF2>[128X[104X
    [4X[25Xgap>[125X [27Xm := [v,v,v];; ConvertToMatrixRep(m,GF(2));; m;[127X[104X
    [4X[28X<a 3x5 matrix over GF2>[128X[104X
    [4X[25Xgap>[125X [27Xv := Z(3)*[1..8];[127X[104X
    [4X[28X[ Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3)^0 ][128X[104X
    [4X[25Xgap>[125X [27XConvertToVectorRep(v);[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27Xm := [v];; ConvertToMatrixRep(m,GF(3));; m;[127X[104X
    [4X[28X[ [ Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3)^0 ] ][128X[104X
    [4X[25Xgap>[125X [27XRepresentationsOfObject(m);[127X[104X
    [4X[28X[ "IsPositionalObjectRep", "Is8BitMatrixRep" ][128X[104X
    [4X[25Xgap>[125X [27Xm := [v,v,v,v];; ConvertToMatrixRep(m,GF(3));; m;[127X[104X
    [4X[28X< mutable compressed matrix 4x8 over GF(3) >[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAll  compressed  matrices  over GF(2) are viewed as [10X<a [3Xn[103X[10Xx[3Xm[103X[10X matrix over GF2>[110X,
  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.[133X
  
  [33X[0;0YMatrices  can  be converted to this special representation via the following
  functions.[133X
  
  [33X[0;0YNote  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.[133X
  
  [33X[0;0YIn  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.[133X
  
  [33X[0;0YThis  is  enforced by the filter [10XIsLockedRepresentationVector[110X. When a vector
  becomes  part of a compressed matrix, this filter is set for it. Assignment,
  [2XUnbind[102X   ([14X21.5-3[114X),   [2XConvertToVectorRep[102X   ([14X23.3-1[114X)   and  [2XConvertToMatrixRep[102X
  ([14X24.14-2[114X) are all prevented from altering a vector with this filter.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xv := [Z(2),Z(2)];; ConvertToVectorRep(v,GF(2));; v;[127X[104X
    [4X[28X<a GF2 vector of length 2>[128X[104X
    [4X[25Xgap>[125X [27Xm := [v,v]; [127X[104X
    [4X[28X[ <a GF2 vector of length 2>, <a GF2 vector of length 2> ][128X[104X
    [4X[25Xgap>[125X [27XConvertToMatrixRep(m,GF(2)); [127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xm2 := [m[1], [Z(4),Z(4)]]; # now try and mix in some GF(4)[127X[104X
    [4X[28X[ <a GF2 vector of length 2>, [ Z(2^2), Z(2^2) ] ][128X[104X
    [4X[25Xgap>[125X [27XConvertToMatrixRep(m2); # but m2[1] is locked[127X[104X
    [4X[28X#I  ConvertToVectorRep: locked vector not converted to different field[128X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xm2 := [ShallowCopy(m[1]), [Z(4),Z(4)]]; # a fresh copy of row 1[127X[104X
    [4X[28X[ <a GF2 vector of length 2>, [ Z(2^2), Z(2^2) ] ][128X[104X
    [4X[25Xgap>[125X [27XConvertToMatrixRep(m2); # now it works[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27Xm2;[127X[104X
    [4X[28X[ [ Z(2)^0, Z(2)^0 ], [ Z(2^2), Z(2^2) ] ][128X[104X
    [4X[25Xgap>[125X [27XRepresentationsOfObject(m2);[127X[104X
    [4X[28X[ "IsPositionalObjectRep", "Is8BitMatrixRep" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YArithmetic  operations  (see [14X21.11[114X  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.[133X
  
  [33X[0;0YThere  are  also two operations that are only available for matrices written
  over finite fields.[133X
  
  [1X24.14-1 ImmutableMatrix[101X
  
  [33X[1;0Y[29X[2XImmutableMatrix[102X( [3Xfield[103X, [3Xmatrix[103X[, [3Xchange[103X] ) [32X operation[133X
  
  [33X[0;0YLet  [3Xmatrix[103X  be  an object for which either [2XIsMatrix[102X ([14X24.2-1[114X) or [2XIsMatrixObj[102X
  ([14X26.2-2[114X)  returns  [9Xtrue[109X.  In the former case, [3Xmatrix[103X is a list of lists, and
  [2XImmutableMatrix[102X  returns  an  immutable  object  for which [2XIsMatrix[102X ([14X24.2-1[114X)
  returns  [9Xtrue[109X  (in  particular  again  a  list  of lists), which is equal to
  [3Xmatrix[103X,  and  which  is  in  the  optimal  (concerning  space  and  runtime)
  representation for matrices defined over [3Xfield[103X, provided that the entries of
  [3Xmatrix[103X  lie  in  [3Xfield[103X.  In  the  latter  case,  [2XImmutableMatrix[102X  returns an
  immutable  object  that is equal to the result of [2XChangedBaseDomain[102X ([14X26.6-3[114X)
  when this is called with [3Xmatrix[103X and [3Xfield[103X.[133X
  
  [33X[0;0YThis  means  that  matrices obtained by several calls of [2XImmutableMatrix[102X for
  the  same  [3Xfield[103X  are  compatible for fast arithmetic without need for field
  conversion.[133X
  
  [33X[0;0YIf the input matrix [3Xmatrix[103X is in [2XIsMatrix[102X ([14X24.2-1[114X) 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 [3Xmatrix[103X is already immutable and
  the  result  of  [2XImmutableMatrix[102X  has  the  same  internal representation as
  [3Xmatrix[103X, the result is not necessarily [13Xidentical[113X to [3Xmatrix[103X.[133X
  
  [33X[0;0YIf  [3Xchange[103X  is  [9Xtrue[109X,  [3Xmatrix[103X  or  its  rows  (if  there are subobjects that
  represent  rows)  may  be changed to become immutable; otherwise the rows of
  [3Xmatrix[103X are copied first.[133X
  
  [1X24.14-2 ConvertToMatrixRep[101X
  
  [33X[1;0Y[29X[2XConvertToMatrixRep[102X( [3Xlist[103X[, [3Xfield[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XConvertToMatrixRep[102X( [3Xlist[103X[, [3Xfieldsize[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XConvertToMatrixRepNC[102X( [3Xlist[103X[, [3Xfield[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XConvertToMatrixRepNC[102X( [3Xlist[103X[, [3Xfieldsize[103X] ) [32X function[133X
  
  [33X[0;0YThis  function is more technical version of [2XImmutableMatrix[102X ([14X24.14-1[114X), 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 [2XImmutableMatrix[102X ([14X24.14-1[114X) 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  [3Xlist[103X,  [2XConvertToMatrixRep[102X converts [3Xlist[103X to an internal matrix
  representation if possible.[133X
  
  [33X[0;0YCalled  with  a  list  [3Xlist[103X  and  a  finite  field [3Xfield[103X, [2XConvertToMatrixRep[102X
  converts  [3Xlist[103X to an internal matrix representation appropriate for a matrix
  over [3Xfield[103X.[133X
  
  [33X[0;0YInstead of a [3Xfield[103X also its size [3Xfieldsize[103X may be given.[133X
  
  [33X[0;0YIt  is  forbidden  to call this function unless all elements of [3Xlist[103X are row
  vectors  with  entries  in  the field [3Xfield[103X. 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 [2XSetAssertionLevel[102X ([14X7.5-1[114X).)[133X
  
  [33X[0;0Y[3Xlist[103X  may  already  be  a  compressed  matrix.  In this case, if no [3Xfield[103X or
  [3Xfieldsize[103X is given, then nothing happens.[133X
  
  [33X[0;0YThe  return  value  is  the  size of the field over which the matrix ends up
  written, if it is written in a compressed representation.[133X
  
  [1X24.14-3 ProjectiveOrder[101X
  
  [33X[1;0Y[29X[2XProjectiveOrder[102X( [3Xmat[103X ) [32X attribute[133X
  
  [33X[0;0YReturns  an  integer  n and a finite field element e such that [3XA[103X^n = eI. [3Xmat[103X
  must be a matrix defined over a finite field.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XProjectiveOrder([[1,4],[5,2]]*Z(11)^0);[127X[104X
    [4X[28X[ 5, Z(11)^5 ][128X[104X
  [4X[32X[104X
  
  [1X24.14-4 SimultaneousEigenvalues[101X
  
  [33X[1;0Y[29X[2XSimultaneousEigenvalues[102X( [3Xmatlist[103X, [3Xexpo[103X ) [32X function[133X
  
  [33X[0;0YThe  matrices  in  [3Xmatlist[103X  must  be  matrices  over GF([3Xq[103X) for some prime [3Xq[103X.
  Together,  they  must generate an abelian p-group of exponent [3Xexpo[103X. Then the
  eigenvalues  of  [3Xmat[103X in the splitting field [10XGF([3Xq[103X[10X^[3Xr[103X[10X)[110X for some [3Xr[103X are powers of
  an   element   [22Xξ[122X   in   the   splitting  field,  which  is  of  order  [3Xexpo[103X.
  [2XSimultaneousEigenvalues[102X  returns  a  matrix  of integers mod [3Xexpo[103X [22X(a_{i,j})[122X,
  such  that  the  power  [22Xξ^{a_{i,j}}[122X  is  an eigenvalue of the [3Xi[103X-th matrix in
  [3Xmatlist[103X  and  the  eigenspaces  of the different matrices to the eigenvalues
  [22Xξ^{a_{i,j}}[122X for fixed [3Xj[103X are equal.[133X
  
  
  [1X24.15 [33X[0;0YInverse and Nullspace of an Integer Matrix Modulo an Ideal[133X[101X
  
  [33X[0;0YThe 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 [22Xp[122X, this is effectively computation in a matrix
  over the prime field in characteristic [22Xp[122X.[133X
  
  [1X24.15-1 InverseMatMod[101X
  
  [33X[1;0Y[29X[2XInverseMatMod[102X( [3Xmat[103X, [3Xobj[103X ) [32X operation[133X
  
  [33X[0;0YFor  a square matrix [3Xmat[103X, [2XInverseMatMod[102X returns a matrix [3Xinv[103X such that [10X[3Xinv[103X[10X *
  [3Xmat[103X[10X[110X is congruent to the identity matrix modulo [3Xobj[103X, if such a matrix exists,
  and [9Xfail[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat:= [ [ 1, 2 ], [ 3, 4 ] ];;  inv:= InverseMatMod( mat, 5 );[127X[104X
    [4X[28X[ [ 3, 1 ], [ 4, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27Xmat * inv;[127X[104X
    [4X[28X[ [ 11, 5 ], [ 25, 11 ] ][128X[104X
  [4X[32X[104X
  
  [1X24.15-2 BasisNullspaceModN[101X
  
  [33X[1;0Y[29X[2XBasisNullspaceModN[102X( [3XM[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0Y[3XM[103X   must   be   a  matrix  of  integers  and  [3Xn[103X  a  positive  integer.  Then
  [2XBasisNullspaceModN[102X  returns  a  set [3XB[103X of vectors such that every vector [3Xv[103X of
  integer  modulo [3Xn[103X satisfying [3Xv[103X [3XM[103X = 0 modulo [3Xn[103X can be expressed by a Z-linear
  combination of elements of [3XB[103X.[133X
  
  [1X24.15-3 NullspaceModQ[101X
  
  [33X[1;0Y[29X[2XNullspaceModQ[102X( [3XM[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNullspaceModN[102X( [3XM[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0Y[3XM[103X  must be a matrix of integers and [3Xn[103X a positive integer. Then [2XNullspaceModN[102X
  returns  the  set  of  all  vectors  of  integers  modulo [3Xn[103X, which solve the
  homogeneous equation system [3Xv[103X [3XM[103X = 0 modulo [3Xn[103X.[133X
  
  [33X[0;0Y[2XNullspaceModQ[102X is a synonym for [2XNullspaceModN[102X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNullspaceModN( [ [ 2 ] ], 8 );[127X[104X
    [4X[28X[ [ 0 ], [ 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XNullspaceModN( [ [ 2, 1 ], [ 0, 2 ] ], 6 );[127X[104X
    [4X[28X[ [ 0, 0 ], [ 0, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27Xmat:= [ [ 1, 3 ], [ 1, 2 ], [ 1, 1 ] ];;[127X[104X
    [4X[25Xgap>[125X [27XNullspaceModN( mat, 5 );[127X[104X
    [4X[28X[ [ 0, 0, 0 ], [ 1, 3, 1 ], [ 2, 1, 2 ], [ 3, 4, 3 ], [ 4, 2, 4 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X24.16 [33X[0;0YSpecial Multiplication Algorithms for Matrices over GF(2)[133X[101X
  
  [33X[0;0YWhen  multiplying two compressed matrices [22XM[122X and [22XN[122X over GF(2) of dimensions [22Xa
  × b[122X and [22Xb × c[122X, where [22Xa[122X, [22Xb[122X and [22Xc[122X are all greater than or equal to 128, [5XGAP[105X by
  default  uses a more sophisticated matrix multiplication algorithm, in which
  linear  combinations  of groups of 8 rows of [22XM[122X 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
  [22X(b+255)/256[122X  sets  of  8  rows  at  once. This number is called the blocking
  level.[133X
  
  [33X[0;0YThese  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.[133X
  
  [33X[0;0YWe  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.[133X
  
  [1X24.16-1 PROD_GF2MAT_GF2MAT_SIMPLE[101X
  
  [33X[1;0Y[29X[2XPROD_GF2MAT_GF2MAT_SIMPLE[102X( [3Xm1[103X, [3Xm2[103X ) [32X function[133X
  
  [33X[0;0YThis   function   performs  the  standard  unblocked  and  ungreased  matrix
  multiplication for matrices of any size.[133X
  
  [1X24.16-2 PROD_GF2MAT_GF2MAT_ADVANCED[101X
  
  [33X[1;0Y[29X[2XPROD_GF2MAT_GF2MAT_ADVANCED[102X( [3Xm1[103X, [3Xm2[103X, [3Xg[103X, [3Xb[103X ) [32X function[133X
  
  [33X[0;0YThis  function  computes  the product of [3Xm1[103X and [3Xm2[103X, which must be compressed
  matrices over GF(2) of compatible dimensions, using level [3Xg[103X grease and level
  [3Xb[103X blocking.[133X
  
  
  [1X24.17 [33X[0;0YBlock Matrices[133X[101X
  
  [33X[0;0YBlock 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.
  [5XGAP[105X uses the representation [10XIsBlockMatrixRep[110X to store block matrices.[133X
  
  [1X24.17-1 AsBlockMatrix[101X
  
  [33X[1;0Y[29X[2XAsBlockMatrix[102X( [3Xm[103X, [3Xnrb[103X, [3Xncb[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  block  matrix with [3Xnrb[103X row blocks and [3Xncb[103X column blocks which is
  equal to the ordinary matrix [3Xm[103X.[133X
  
  [1X24.17-2 BlockMatrix[101X
  
  [33X[1;0Y[29X[2XBlockMatrix[102X( [3Xblocks[103X, [3Xnrb[103X, [3Xncb[103X[, [3Xrpb[103X, [3Xcpb[103X, [3Xzero[103X] ) [32X function[133X
  
  [33X[0;0Y[2XBlockMatrix[102X  returns  an  immutable  matrix  in  the  sparse  representation
  [10XIsBlockMatrixRep[110X.  The  nonzero  blocks  are described by the list [3Xblocks[103X of
  triples  [22X[  [3Xi[103X,  [3Xj[103X, M(i,j) ][122X each consisting of a matrix [22XM(i,j)[122X and its block
  coordinates  in the block matrix to be constructed. All matrices [22XM(i,j)[122X must
  have  the  same  dimensions. As usual the first coordinate specifies the row
  and  the  second one the column. The resulting matrix has [3Xnrb[103X row blocks and
  [3Xncb[103X column blocks.[133X
  
  [33X[0;0YIf  [3Xblocks[103X  is  empty  (i.e.,  if  the  matrix  is  a  zero matrix) then the
  dimensions  of  the  blocks  must  be  entered  as [3Xrpb[103X and [3Xcpb[103X, and the zero
  element as [3Xzero[103X.[133X
  
  [33X[0;0YNote  that  all  blocks  must  be  ordinary  matrices  (see [2XIsOrdinaryMatrix[102X
  ([14X24.2-2[114X)), and also the block matrix is an ordinary matrix.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XM := BlockMatrix([[1,1,[[1, 2],[ 3, 4]]],[127X[104X
    [4X[25X>[125X [27X                     [1,2,[[9,10],[11,12]]],[127X[104X
    [4X[25X>[125X [27X                     [2,2,[[5, 6],[ 7, 8]]]],2,2);[127X[104X
    [4X[28X<block matrix of dimensions (2*2)x(2*2)>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(M);[127X[104X
    [4X[28X[ [   1,   2,   9,  10 ],[128X[104X
    [4X[28X  [   3,   4,  11,  12 ],[128X[104X
    [4X[28X  [   0,   0,   5,   6 ],[128X[104X
    [4X[28X  [   0,   0,   7,   8 ] ][128X[104X
  [4X[32X[104X
  
  [1X24.17-3 MatrixByBlockMatrix[101X
  
  [33X[1;0Y[29X[2XMatrixByBlockMatrix[102X( [3Xblockmat[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns a plain ordinary matrix that is equal to the block matrix [3Xblockmat[103X.[133X
  
  
  [1X24.18 [33X[0;0YLinear Programming[133X[101X
  
  [1X24.18-1 SimplexMethod[101X
  
  [33X[1;0Y[29X[2XSimplexMethod[102X( [3XA[103X, [3Xb[103X, [3Xc[103X ) [32X function[133X
  
  [33X[0;0YFind  a  rational  vector  [3Xx[103X  that  maximizes [22X[3Xx[103X⋅[3Xc[103X[122X, subject to the constraint
  [22X[3XA[103X⋅[3Xx[103X≤[3Xb[103X[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA:=[[3,1,1,4],[1,-3,2,3],[2,1,3,-1]];;[127X[104X
    [4X[25Xgap>[125X [27Xb:=[12,7,10];;c:=[2,4,3,1];;[127X[104X
    [4X[25Xgap>[125X [27XSimplexMethod(A,b,c);[127X[104X
    [4X[28X[ [ 0, 52/5, 0, 2/5 ], 42 ][128X[104X
  [4X[32X[104X
  
