  
  [1X5 [33X[0;0YMatrices[133X[101X
  
  
  [1X5.1 [33X[0;0YMatrices: Category and Representations[133X[101X
  
  [1X5.1-1 IsHomalgMatrix[101X
  
  [33X[1;0Y[29X[2XIsHomalgMatrix[102X( [3XA[103X ) [32X Category[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe [5XGAP[105X category of [5Xhomalg[105X matrices.[133X
  
  [4X[32X  Code  [32X[104X
    [4XDeclareCategory( "IsHomalgMatrix",[104X
    [4X        IsMatrixObj and[104X
    [4X        IsAttributeStoringRep );[104X
  [4X[32X[104X
  
  [1X5.1-2 IsHomalgInternalMatrixRep[101X
  
  [33X[1;0Y[29X[2XIsHomalgInternalMatrixRep[102X( [3XA[103X ) [32X Representation[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe internal representation of [5Xhomalg[105X matrices.[133X
  
  [33X[0;0Y(It is a representation of the [5XGAP[105X category [2XIsHomalgMatrix[102X ([14X5.1-1[114X).)[133X
  
  
  [1X5.2 [33X[0;0YMatrices: Constructors[133X[101X
  
  [1X5.2-1 HomalgInitialMatrix[101X
  
  [33X[1;0Y[29X[2XHomalgInitialMatrix[102X( [3Xm[103X, [3Xn[103X, [3XR[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YA mutable unevaluated initial [22X[3Xm[103X × [3Xn[103X[122X [5Xhomalg[105X matrix filled with zeros over the
  [5Xhomalg[105X  ring  [3XR[103X.  This  construction is useful in case one wants to define a
  matrix  by  assigning  its  nonzero  entries.  The  property [2XIsInitialMatrix[102X
  ([14X5.3-26[114X)  is  reset  as  soon  as  the  matrix  is  evaluated.  New computed
  properties  or attributes of the matrix won't be cached, until the matrix is
  explicitly    made    immutable   using   (-->   [2XMakeImmutable[102X   ([14XReference:
  MakeImmutable[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZZ := HomalgRingOfIntegers( );[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27Xz := HomalgInitialMatrix( 2, 3, ZZ );[127X[104X
    [4X[28X<An initial 2 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XHasIsZero( z );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsZero( z );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xz;[127X[104X
    [4X[28X<A 2 x 3 mutable matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XHasIsZero( z );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn := HomalgInitialMatrix( 2, 3, ZZ );[127X[104X
    [4X[28X<An initial 2 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27Xn[ 1, 1 ] := "1";;[127X[104X
    [4X[25Xgap>[125X [27Xn[ 2, 3 ] := "1";;[127X[104X
    [4X[25Xgap>[125X [27XMakeImmutable( n );[127X[104X
    [4X[28X<A 2 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( n );[127X[104X
    [4X[28X[ [  1,  0,  0 ],[128X[104X
    [4X[28X  [  0,  0,  1 ] ][128X[104X
    [4X[25Xgap>[125X [27XIsZero( n );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xn;[127X[104X
    [4X[28X<A non-zero 2 x 3 matrix over an internal ring>[128X[104X
  [4X[32X[104X
  
  [1X5.2-2 HomalgInitialIdentityMatrix[101X
  
  [33X[1;0Y[29X[2XHomalgInitialIdentityMatrix[102X( [3Xm[103X, [3XR[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YA mutable unevaluated initial [22X[3Xm[103X × [3Xm[103X[122X [5Xhomalg[105X quadratic matrix with ones on the
  diagonal  over  the  [5Xhomalg[105X  ring [3XR[103X. This construction is useful in case one
  wants  to  define an elementary matrix by assigning its off-diagonal nonzero
  entries.  The  property [2XIsInitialIdentityMatrix[102X ([14X5.3-27[114X) is reset as soon as
  the matrix is evaluated. New computed properties or attributes of the matrix
  won't  be  cached,  until the matrix is explicitly made immutable using (-->
  [2XMakeImmutable[102X ([14XReference: MakeImmutable[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZZ := HomalgRingOfIntegers( );[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27Xid := HomalgInitialIdentityMatrix( 3, ZZ );[127X[104X
    [4X[28X<An initial identity 3 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XHasIsOne( id );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsOne( id );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xid;[127X[104X
    [4X[28X<A 3 x 3 mutable matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XHasIsOne( id );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xe := HomalgInitialIdentityMatrix( 3, ZZ );[127X[104X
    [4X[28X<An initial identity 3 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27Xe[ 1, 2 ] := "1";;[127X[104X
    [4X[25Xgap>[125X [27Xe[ 2, 1 ] := "-1";;[127X[104X
    [4X[25Xgap>[125X [27XMakeImmutable( e );[127X[104X
    [4X[28X<A 3 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( e );[127X[104X
    [4X[28X[ [   1,   1,   0 ],[128X[104X
    [4X[28X  [  -1,   1,   0 ],[128X[104X
    [4X[28X  [   0,   0,   1 ] ][128X[104X
    [4X[25Xgap>[125X [27XIsOne( e );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xe;[127X[104X
    [4X[28X<A 3 x 3 matrix over an internal ring>[128X[104X
  [4X[32X[104X
  
  [1X5.2-3 HomalgZeroMatrix[101X
  
  [33X[1;0Y[29X[2XHomalgZeroMatrix[102X( [3Xm[103X, [3Xn[103X, [3XR[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YAn immutable unevaluated [22X[3Xm[103X × [3Xn[103X[122X [5Xhomalg[105X zero matrix over the [5Xhomalg[105X ring [3XR[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZZ := HomalgRingOfIntegers( );[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27Xz := HomalgZeroMatrix( 2, 3, ZZ );[127X[104X
    [4X[28X<An unevaluated 2 x 3 zero matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( z );[127X[104X
    [4X[28X[ [  0,  0,  0 ],[128X[104X
    [4X[28X  [  0,  0,  0 ] ][128X[104X
    [4X[25Xgap>[125X [27Xz;[127X[104X
    [4X[28X<A 2 x 3 zero matrix over an internal ring>[128X[104X
  [4X[32X[104X
  
  [1X5.2-4 HomalgIdentityMatrix[101X
  
  [33X[1;0Y[29X[2XHomalgIdentityMatrix[102X( [3Xm[103X, [3XR[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YAn  immutable  unevaluated [22X[3Xm[103X × [3Xm[103X[122X [5Xhomalg[105X identity matrix over the [5Xhomalg[105X ring
  [3XR[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZZ := HomalgRingOfIntegers( );[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27Xid := HomalgIdentityMatrix( 3, ZZ );[127X[104X
    [4X[28X<An unevaluated 3 x 3 identity matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( id );[127X[104X
    [4X[28X[ [  1,  0,  0 ],[128X[104X
    [4X[28X  [  0,  1,  0 ],[128X[104X
    [4X[28X  [  0,  0,  1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xid;[127X[104X
    [4X[28X<A 3 x 3 identity matrix over an internal ring>[128X[104X
  [4X[32X[104X
  
  [1X5.2-5 HomalgVoidMatrix[101X
  
  [33X[1;0Y[29X[2XHomalgVoidMatrix[102X( [[3Xm[103X, ][[3Xn[103X, ][3XR[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YA void [22X[3Xm[103X × [3Xn[103X[122X [5Xhomalg[105X matrix.[133X
  
  [1X5.2-6 HomalgMatrix[101X
  
  [33X[1;0Y[29X[2XHomalgMatrix[102X( [3Xllist[103X, [3XR[103X ) [32X function[133X
  [33X[1;0Y[29X[2XHomalgMatrix[102X( [3Xllist[103X, [3Xm[103X, [3Xn[103X, [3XR[103X ) [32X function[133X
  [33X[1;0Y[29X[2XHomalgMatrix[102X( [3Xlist[103X, [3Xm[103X, [3Xn[103X, [3XR[103X ) [32X function[133X
  [33X[1;0Y[29X[2XHomalgMatrix[102X( [3Xstr_llist[103X, [3XR[103X ) [32X function[133X
  [33X[1;0Y[29X[2XHomalgMatrix[102X( [3Xstr_list[103X, [3Xm[103X, [3Xn[103X, [3XR[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YAn immutable evaluated [22X[3Xm[103X × [3Xn[103X[122X [5Xhomalg[105X matrix over the [5Xhomalg[105X ring [3XR[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZZ := HomalgRingOfIntegers( );[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27Xm := HomalgMatrix( [ [ 1, 2, 3 ], [ 4, 5, 6 ] ], ZZ );[127X[104X
    [4X[28X<A 2 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( m );[127X[104X
    [4X[28X[ [  1,  2,  3 ],[128X[104X
    [4X[28X  [  4,  5,  6 ] ][128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm := HomalgMatrix( [ [ 1, 2, 3 ], [ 4, 5, 6 ] ], 2, 3, ZZ );[127X[104X
    [4X[28X<A 2 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( m );[127X[104X
    [4X[28X[ [  1,  2,  3 ],[128X[104X
    [4X[28X  [  4,  5,  6 ] ][128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm := HomalgMatrix( [ 1, 2, 3,   4, 5, 6 ], 2, 3, ZZ );[127X[104X
    [4X[28X<A 2 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( m );[127X[104X
    [4X[28X[ [  1,  2,  3 ],[128X[104X
    [4X[28X  [  4,  5,  6 ] ][128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm := HomalgMatrix( "[ [ 1, 2, 3 ], [ 4, 5, 6 ] ]", ZZ );[127X[104X
    [4X[28X<A 2 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( m );[127X[104X
    [4X[28X[ [  1,  2,  3 ],[128X[104X
    [4X[28X  [  4,  5,  6 ] ][128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm := HomalgMatrix( "[ [ 1, 2, 3 ], [ 4, 5, 6 ] ]", 2, 3, ZZ );[127X[104X
    [4X[28X<A 2 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( m );[127X[104X
    [4X[28X[ [  1,  2,  3 ],[128X[104X
    [4X[28X  [  4,  5,  6 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt  is  nevertheless  recommended to use the following form to create [5Xhomalg[105X
  matrices.  This  form  can  also  be used to define external matrices. Since
  whitespaces  (-->  [14XReference:  Whitespaces[114X) are ignored, they can be used as
  optical delimiters:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm := HomalgMatrix( "[ 1, 2, 3,   4, 5, 6 ]", 2, 3, ZZ );[127X[104X
    [4X[28X<A 2 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( m );[127X[104X
    [4X[28X[ [  1,  2,  3 ],[128X[104X
    [4X[28X  [  4,  5,  6 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YOne  can  split  the  input  string  over  several lines using the backslash
  character '\' to end each line[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm := HomalgMatrix( "[ \[127X[104X
    [4X[25X>[125X [27X1, 2, 3, \[127X[104X
    [4X[25X>[125X [27X4, 5, 6  \[127X[104X
    [4X[25X>[125X [27X]", 2, 3, ZZ );[127X[104X
    [4X[28X<A 2 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( m );[127X[104X
    [4X[28X[ [  1,  2,  3 ],[128X[104X
    [4X[28X  [  4,  5,  6 ] ][128X[104X
  [4X[32X[104X
  
  [1X5.2-7 HomalgMatrixListList[101X
  
  [33X[1;0Y[29X[2XHomalgMatrixListList[102X( [3Xllist[103X, [3Xm[103X, [3Xn[103X, [3XR[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YSpecial case of [2XHomalgMatrix[102X ([14X5.2-6[114X).[133X
  
  [1X5.2-8 HomalgRowVector[101X
  
  [33X[1;0Y[29X[2XHomalgRowVector[102X( [3Xentries[103X, [3Xnr_cols[103X, [3XR[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YSpecial case of [2XHomalgMatrix[102X ([14X5.2-6[114X) for matrices with a single row. [3Xentries[103X
  must be a list of ring elements.[133X
  
  [1X5.2-9 HomalgColumnVector[101X
  
  [33X[1;0Y[29X[2XHomalgColumnVector[102X( [3Xentries[103X, [3Xnr_rows[103X, [3XR[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YSpecial  case  of  [2XHomalgMatrix[102X  ([14X5.2-6[114X)  for matrices with a single column.
  [3Xentries[103X must be a list of ring elements.[133X
  
  [1X5.2-10 HomalgDiagonalMatrix[101X
  
  [33X[1;0Y[29X[2XHomalgDiagonalMatrix[102X( [3Xdiag[103X, [3XR[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YAn  immutable unevaluated diagonal [5Xhomalg[105X matrix over the [5Xhomalg[105X ring [3XR[103X. The
  diagonal consists of the entries of the list [3Xdiag[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZZ := HomalgRingOfIntegers( );[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27Xd := HomalgDiagonalMatrix( [ 1, 2, 3 ], ZZ );[127X[104X
    [4X[28X<An unevaluated diagonal 3 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( d );[127X[104X
    [4X[28X[ [  1,  0,  0 ],[128X[104X
    [4X[28X  [  0,  2,  0 ],[128X[104X
    [4X[28X  [  0,  0,  3 ] ][128X[104X
    [4X[25Xgap>[125X [27Xd;[127X[104X
    [4X[28X<A diagonal 3 x 3 matrix over an internal ring>[128X[104X
  [4X[32X[104X
  
  [1X5.2-11 \*[101X
  
  [33X[1;0Y[29X[2X\*[102X( [3XR[103X, [3Xmat[103X ) [32X operation[133X
  [33X[1;0Y[29X[2X\*[102X( [3Xmat[103X, [3XR[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YAn  immutable evaluated [5Xhomalg[105X matrix over the [5Xhomalg[105X ring [3XR[103X having the same
  entries as the matrix [3Xmat[103X. Syntax: [3XR[103X [10X*[110X [3Xmat[103X or [3Xmat[103X [10X*[110X [3XR[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZZ := HomalgRingOfIntegers( );[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27XZ4 := ZZ / 4;[127X[104X
    [4X[28XZ/( 4 )[128X[104X
    [4X[25Xgap>[125X [27XDisplay( Z4 );[127X[104X
    [4X[28X<A residue class ring>[128X[104X
    [4X[25Xgap>[125X [27Xd := HomalgDiagonalMatrix( [ 2 .. 4 ], ZZ );[127X[104X
    [4X[28X<An unevaluated diagonal 3 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27Xd2 := Z4 * d; ## or d2 := d * Z4;[127X[104X
    [4X[28X<A 3 x 3 matrix over a residue class ring>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( d2 );[127X[104X
    [4X[28X[ [  2,  0,  0 ],[128X[104X
    [4X[28X  [  0,  3,  0 ],[128X[104X
    [4X[28X  [  0,  0,  4 ] ][128X[104X
    [4X[28X[128X[104X
    [4X[28Xmodulo [ 4 ][128X[104X
    [4X[25Xgap>[125X [27Xd;[127X[104X
    [4X[28X<A diagonal 3 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XZeroRows( d );[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27XZeroRows( d2 );[127X[104X
    [4X[28X[ 3 ][128X[104X
    [4X[25Xgap>[125X [27Xd;[127X[104X
    [4X[28X<A non-zero diagonal 3 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27Xd2;[127X[104X
    [4X[28X<A non-zero 3 x 3 matrix over a residue class ring>[128X[104X
  [4X[32X[104X
  
  [1X5.2-12 CoercedMatrix[101X
  
  [33X[1;0Y[29X[2XCoercedMatrix[102X( [3Xring_from[103X, [3Xring_to[103X, [3Xmat[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XCoercedMatrix[102X( [3Xring_to[103X, [3Xmat[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YA  copy  of  the  [5Xhomalg[105X matrix [3Xmat[103X with [5Xhomalg[105X ring [3Xring_from[103X in the [5Xhomalg[105X
  ring [3Xring_to[103X.[133X
  
  [33X[0;0Y(for the installed standard method see [2XEval[102X ([14XC.4-9[114X))[133X
  
  
  [1X5.3 [33X[0;0YMatrices: Properties[133X[101X
  
  [1X5.3-1 IsZero[101X
  
  [33X[1;0Y[29X[2XIsZero[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck  if  the  [5Xhomalg[105X  matrix  [3XA[103X  is  a  zero  matrix, taking possible ring
  relations into account.[133X
  
  [33X[0;0Y(for the installed standard method see [2XIsZeroMatrix[102X ([14XB.1-20[114X))[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZZ := HomalgRingOfIntegers( );[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27XA := HomalgMatrix( "[ 2 ]", ZZ );[127X[104X
    [4X[28X<A 1 x 1 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XZ2 := ZZ / 2;[127X[104X
    [4X[28XZ/( 2 )[128X[104X
    [4X[25Xgap>[125X [27XA := Z2 * A;[127X[104X
    [4X[28X<A 1 x 1 matrix over a residue class ring>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( A );[127X[104X
    [4X[28X[ [  2 ] ][128X[104X
    [4X[28X[128X[104X
    [4X[28Xmodulo [ 2 ][128X[104X
    [4X[25Xgap>[125X [27XIsZero( A );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X5.3-2 IsOne[101X
  
  [33X[1;0Y[29X[2XIsOne[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck  if  the  [5Xhomalg[105X  matrix [3XA[103X is an identity matrix, taking possible ring
  relations into account.[133X
  
  [33X[0;0Y(for the installed standard method see [2XIsIdentityMatrix[102X ([14XB.2-2[114X))[133X
  
  [1X5.3-3 IsUnitFree[101X
  
  [33X[1;0Y[29X[2XIsUnitFree[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-4 IsPermutationMatrix[101X
  
  [33X[1;0Y[29X[2XIsPermutationMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-5 IsSpecialSubidentityMatrix[101X
  
  [33X[1;0Y[29X[2XIsSpecialSubidentityMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-6 IsSubidentityMatrix[101X
  
  [33X[1;0Y[29X[2XIsSubidentityMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-7 IsLeftRegular[101X
  
  [33X[1;0Y[29X[2XIsLeftRegular[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-8 IsRightRegular[101X
  
  [33X[1;0Y[29X[2XIsRightRegular[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-9 IsInvertibleMatrix[101X
  
  [33X[1;0Y[29X[2XIsInvertibleMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-10 IsLeftInvertibleMatrix[101X
  
  [33X[1;0Y[29X[2XIsLeftInvertibleMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-11 IsRightInvertibleMatrix[101X
  
  [33X[1;0Y[29X[2XIsRightInvertibleMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-12 IsEmptyMatrix[101X
  
  [33X[1;0Y[29X[2XIsEmptyMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-13 IsDiagonalMatrix[101X
  
  [33X[1;0Y[29X[2XIsDiagonalMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck  if  the  [5Xhomalg[105X  matrix  [3XA[103X is a diagonal matrix, taking possible ring
  relations into account.[133X
  
  [33X[0;0Y(for the installed standard method see [2XIsDiagonalMatrix[102X ([14XB.2-3[114X))[133X
  
  [1X5.3-14 IsScalarlMatrix[101X
  
  [33X[1;0Y[29X[2XIsScalarlMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-15 IsUpperTriangularMatrix[101X
  
  [33X[1;0Y[29X[2XIsUpperTriangularMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-16 IsLowerTriangularMatrix[101X
  
  [33X[1;0Y[29X[2XIsLowerTriangularMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-17 IsStrictUpperTriangularMatrix[101X
  
  [33X[1;0Y[29X[2XIsStrictUpperTriangularMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-18 IsStrictLowerTriangularMatrix[101X
  
  [33X[1;0Y[29X[2XIsStrictLowerTriangularMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-19 IsUpperStairCaseMatrix[101X
  
  [33X[1;0Y[29X[2XIsUpperStairCaseMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-20 IsLowerStairCaseMatrix[101X
  
  [33X[1;0Y[29X[2XIsLowerStairCaseMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-21 IsTriangularMatrix[101X
  
  [33X[1;0Y[29X[2XIsTriangularMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-22 IsBasisOfRowsMatrix[101X
  
  [33X[1;0Y[29X[2XIsBasisOfRowsMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-23 IsBasisOfColumnsMatrix[101X
  
  [33X[1;0Y[29X[2XIsBasisOfColumnsMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-24 IsReducedBasisOfRowsMatrix[101X
  
  [33X[1;0Y[29X[2XIsReducedBasisOfRowsMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-25 IsReducedBasisOfColumnsMatrix[101X
  
  [33X[1;0Y[29X[2XIsReducedBasisOfColumnsMatrix[102X( [3XA[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-26 IsInitialMatrix[101X
  
  [33X[1;0Y[29X[2XIsInitialMatrix[102X( [3XA[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-27 IsInitialIdentityMatrix[101X
  
  [33X[1;0Y[29X[2XIsInitialIdentityMatrix[102X( [3XA[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  [1X5.3-28 IsVoidMatrix[101X
  
  [33X[1;0Y[29X[2XIsVoidMatrix[102X( [3XA[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  
  [1X5.4 [33X[0;0YMatrices: Attributes[133X[101X
  
  [1X5.4-1 NrRows[101X
  
  [33X[1;0Y[29X[2XNrRows[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya nonnegative integer[133X
  
  [33X[0;0YThe number of rows of the matrix [3XA[103X.[133X
  
  [33X[0;0Y(for the installed standard method see [2XNrRows[102X ([14XB.1-21[114X))[133X
  
  [1X5.4-2 NrColumns[101X
  
  [33X[1;0Y[29X[2XNrColumns[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya nonnegative integer[133X
  
  [33X[0;0YThe number of columns of the matrix [3XA[103X.[133X
  
  [33X[0;0Y(for the installed standard method see [2XNrColumns[102X ([14XB.1-22[114X))[133X
  
  [1X5.4-3 DeterminantMat[101X
  
  [33X[1;0Y[29X[2XDeterminantMat[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya ring element[133X
  
  [33X[0;0YThe determinant of the quadratic matrix [3XA[103X.[133X
  
  [33X[0;0YYou can invoke it with [10XDeterminant[110X( [3XA[103X ).[133X
  
  [33X[0;0Y(for the installed standard method see [2XDeterminant[102X ([14XB.1-23[114X))[133X
  
  [1X5.4-4 ZeroRows[101X
  
  [33X[1;0Y[29X[2XZeroRows[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya (possibly empty) list of positive integers[133X
  
  [33X[0;0YThe list of zero rows of the matrix [3XA[103X.[133X
  
  [33X[0;0Y(for the installed standard method see [2XZeroRows[102X ([14XB.2-4[114X))[133X
  
  [1X5.4-5 ZeroColumns[101X
  
  [33X[1;0Y[29X[2XZeroColumns[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya (possibly empty) list of positive integers[133X
  
  [33X[0;0YThe list of zero columns of the matrix [3XA[103X.[133X
  
  [33X[0;0Y(for the installed standard method see [2XZeroColumns[102X ([14XB.2-5[114X))[133X
  
  [1X5.4-6 NonZeroRows[101X
  
  [33X[1;0Y[29X[2XNonZeroRows[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya (possibly empty) list of positive integers[133X
  
  [33X[0;0YThe list of nonzero rows of the matrix [3XA[103X.[133X
  
  [1X5.4-7 NonZeroColumns[101X
  
  [33X[1;0Y[29X[2XNonZeroColumns[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya (possibly empty) list of positive integers[133X
  
  [33X[0;0YThe list of nonzero columns of the matrix [3XA[103X.[133X
  
  [1X5.4-8 PositionOfFirstNonZeroEntryPerRow[101X
  
  [33X[1;0Y[29X[2XPositionOfFirstNonZeroEntryPerRow[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list of nonnegative integers[133X
  
  [33X[0;0YThe  list  of  positions of the first nonzero entry per row of the matrix [3XA[103X,
  else zero.[133X
  
  [1X5.4-9 PositionOfFirstNonZeroEntryPerColumn[101X
  
  [33X[1;0Y[29X[2XPositionOfFirstNonZeroEntryPerColumn[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list of nonnegative integers[133X
  
  [33X[0;0YThe list of positions of the first nonzero entry per column of the matrix [3XA[103X,
  else zero.[133X
  
  [1X5.4-10 RowRankOfMatrix[101X
  
  [33X[1;0Y[29X[2XRowRankOfMatrix[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya nonnegative integer[133X
  
  [33X[0;0YThe row rank of the matrix [3XA[103X.[133X
  
  [1X5.4-11 ColumnRankOfMatrix[101X
  
  [33X[1;0Y[29X[2XColumnRankOfMatrix[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya nonnegative integer[133X
  
  [33X[0;0YThe column rank of the matrix [3XA[103X.[133X
  
  [1X5.4-12 LeftInverse[101X
  
  [33X[1;0Y[29X[2XLeftInverse[102X( [3XM[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YA  left  inverse  [22XC[122X of the matrix [3XM[103X. If no left inverse exists then [10Xfalse[110X is
  returned. (--> [2XRightDivide[102X ([14X5.5-51[114X))[133X
  
  [33X[0;0Y(for the installed standard method see [2XLeftInverse[102X ([14X5.5-2[114X))[133X
  
  [1X5.4-13 RightInverse[101X
  
  [33X[1;0Y[29X[2XRightInverse[102X( [3XM[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YA  right inverse [22XC[122X of the matrix [3XM[103X. If no right inverse exists then [10Xfalse[110X is
  returned. (--> [2XLeftDivide[102X ([14X5.5-52[114X))[133X
  
  [33X[0;0Y(for the installed standard method see [2XRightInverse[102X ([14X5.5-3[114X))[133X
  
  [1X5.4-14 CoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries[101X
  
  [33X[1;0Y[29X[2XCoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list of integers[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix (row convention).[133X
  
  [1X5.4-15 CoefficientsOfNumeratorOfHilbertPoincareSeries[101X
  
  [33X[1;0Y[29X[2XCoefficientsOfNumeratorOfHilbertPoincareSeries[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list of integers[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix (row convention).[133X
  
  [1X5.4-16 UnreducedNumeratorOfHilbertPoincareSeries[101X
  
  [33X[1;0Y[29X[2XUnreducedNumeratorOfHilbertPoincareSeries[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya univariate polynomial with rational coefficients[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix (row convention).[133X
  
  [1X5.4-17 NumeratorOfHilbertPoincareSeries[101X
  
  [33X[1;0Y[29X[2XNumeratorOfHilbertPoincareSeries[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya univariate polynomial with rational coefficients[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix (row convention).[133X
  
  [1X5.4-18 HilbertPoincareSeries[101X
  
  [33X[1;0Y[29X[2XHilbertPoincareSeries[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya univariate rational function with rational coefficients[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix (row convention).[133X
  
  [1X5.4-19 HilbertPolynomial[101X
  
  [33X[1;0Y[29X[2XHilbertPolynomial[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya univariate polynomial with rational coefficients[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix (row convention).[133X
  
  [1X5.4-20 AffineDimension[101X
  
  [33X[1;0Y[29X[2XAffineDimension[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix (row convention).[133X
  
  [1X5.4-21 AffineDegree[101X
  
  [33X[1;0Y[29X[2XAffineDegree[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya nonnegative integer[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix (row convention).[133X
  
  [1X5.4-22 ProjectiveDegree[101X
  
  [33X[1;0Y[29X[2XProjectiveDegree[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya nonnegative integer[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix (row convention).[133X
  
  [1X5.4-23 ConstantTermOfHilbertPolynomialn[101X
  
  [33X[1;0Y[29X[2XConstantTermOfHilbertPolynomialn[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix (row convention).[133X
  
  [1X5.4-24 MatrixOfSymbols[101X
  
  [33X[1;0Y[29X[2XMatrixOfSymbols[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0Y[3XA[103X is a [5Xhomalg[105X matrix.[133X
  
  
  [1X5.5 [33X[0;0YMatrices: Operations and Functions[133X[101X
  
  [1X5.5-1 HomalgRing[101X
  
  [33X[1;0Y[29X[2XHomalgRing[102X( [3Xmat[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X ring[133X
  
  [33X[0;0YThe [5Xhomalg[105X ring of the [5Xhomalg[105X matrix [3Xmat[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZZ := HomalgRingOfIntegers( );[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27Xd := HomalgDiagonalMatrix( [ 2 .. 4 ], ZZ );[127X[104X
    [4X[28X<An unevaluated diagonal 3 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XR := HomalgRing( d );[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27XIsIdenticalObj( R, ZZ );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X5.5-2 LeftInverse[101X
  
  [33X[1;0Y[29X[2XLeftInverse[102X( [3XRI[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix or fail[133X
  
  [33X[0;0YThe  left  inverse  of  the matrix [3XRI[103X. The lazy version of this operation is
  [2XLeftInverseLazy[102X ([14X5.5-4[114X). (--> [2XRightDivide[102X ([14X5.5-51[114X))[133X
  
  [4X[32X  Code  [32X[104X
    [4XInstallMethod( LeftInverse,[104X
    [4X        "for homalg matrices",[104X
    [4X        [ IsHomalgMatrix ],[104X
    [4X        [104X
    [4X  function( RI )[104X
    [4X    local Id, LI;[104X
    [4X    [104X
    [4X    Id := HomalgIdentityMatrix( NrColumns( RI ), HomalgRing( RI ) );[104X
    [4X    [104X
    [4X    LI := RightDivide( Id, RI ); ## ( cf. [BR08, Subsection 3.1.3] )[104X
    [4X    [104X
    [4X    ## CAUTION: for the following SetXXX RightDivide is assumed[104X
    [4X    ## NOT to be lazy evaluated!!![104X
    [4X    [104X
    [4X    SetIsLeftInvertibleMatrix( RI, IsHomalgMatrix( LI ) );[104X
    [4X    [104X
    [4X    if IsBool( LI ) then[104X
    [4X        return fail;[104X
    [4X    fi;[104X
    [4X    [104X
    [4X    if HasIsInvertibleMatrix( RI ) and IsInvertibleMatrix( RI ) then[104X
    [4X        SetIsInvertibleMatrix( LI, true );[104X
    [4X    else[104X
    [4X        SetIsRightInvertibleMatrix( LI, true );[104X
    [4X    fi;[104X
    [4X    [104X
    [4X    SetRightInverse( LI, RI );[104X
    [4X    [104X
    [4X    SetNrColumns( LI, NrRows( RI ) );[104X
    [4X    [104X
    [4X    if NrRows( RI ) = NrColumns( RI ) then[104X
    [4X        ## a left inverse of a ring element is unique[104X
    [4X        ## and coincides with the right inverse[104X
    [4X        SetRightInverse( RI, LI );[104X
    [4X        SetLeftInverse( LI, RI );[104X
    [4X    fi;[104X
    [4X    [104X
    [4X    return LI;[104X
    [4X    [104X
    [4Xend );[104X
  [4X[32X[104X
  
  [1X5.5-3 RightInverse[101X
  
  [33X[1;0Y[29X[2XRightInverse[102X( [3XLI[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix or fail[133X
  
  [33X[0;0YThe  right  inverse  of the matrix [3XLI[103X. The lazy version of this operation is
  [2XRightInverseLazy[102X ([14X5.5-5[114X). (--> [2XLeftDivide[102X ([14X5.5-52[114X))[133X
  
  [4X[32X  Code  [32X[104X
    [4XInstallMethod( RightInverse,[104X
    [4X        "for homalg matrices",[104X
    [4X        [ IsHomalgMatrix ],[104X
    [4X        [104X
    [4X  function( LI )[104X
    [4X    local Id, RI;[104X
    [4X    [104X
    [4X    Id := HomalgIdentityMatrix( NrRows( LI ), HomalgRing( LI ) );[104X
    [4X    [104X
    [4X    RI := LeftDivide( LI, Id ); ## ( cf. [BR08, Subsection 3.1.3] )[104X
    [4X    [104X
    [4X    ## CAUTION: for the following SetXXX LeftDivide is assumed[104X
    [4X    ## NOT to be lazy evaluated!!![104X
    [4X    [104X
    [4X    SetIsRightInvertibleMatrix( LI, IsHomalgMatrix( RI ) );[104X
    [4X    [104X
    [4X    if IsBool( RI ) then[104X
    [4X        return fail;[104X
    [4X    fi;[104X
    [4X    [104X
    [4X    if HasIsInvertibleMatrix( LI ) and IsInvertibleMatrix( LI ) then[104X
    [4X        SetIsInvertibleMatrix( RI, true );[104X
    [4X    else[104X
    [4X        SetIsLeftInvertibleMatrix( RI, true );[104X
    [4X    fi;[104X
    [4X    [104X
    [4X    SetLeftInverse( RI, LI );[104X
    [4X    [104X
    [4X    SetNrRows( RI, NrColumns( LI ) );[104X
    [4X    [104X
    [4X    if NrRows( LI ) = NrColumns( LI ) then[104X
    [4X        ## a right inverse of a ring element is unique[104X
    [4X        ## and coincides with the left inverse[104X
    [4X        SetLeftInverse( LI, RI );[104X
    [4X        SetRightInverse( RI, LI );[104X
    [4X    fi;[104X
    [4X    [104X
    [4X    return RI;[104X
    [4X    [104X
    [4Xend );[104X
  [4X[32X[104X
  
  [1X5.5-4 LeftInverseLazy[101X
  
  [33X[1;0Y[29X[2XLeftInverseLazy[102X( [3XM[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YA  lazy  evaluated left inverse [22XC[122X of the matrix [3XM[103X. If no left inverse exists
  then [10XEval[110X( [3XC[103X ) will issue an error.[133X
  
  [33X[0;0Y(for the installed standard method see [2XEval[102X ([14XC.4-5[114X))[133X
  
  [1X5.5-5 RightInverseLazy[101X
  
  [33X[1;0Y[29X[2XRightInverseLazy[102X( [3XM[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YA lazy evaluated right inverse [22XC[122X of the matrix [3XM[103X. If no right inverse exists
  then [10XEval[110X( [3XC[103X ) will issue an error.[133X
  
  [33X[0;0Y(for the installed standard method see [2XEval[102X ([14XC.4-6[114X))[133X
  
  [1X5.5-6 Involution[101X
  
  [33X[1;0Y[29X[2XInvolution[102X( [3XM[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YThe  twisted  transpose  of  the  [5Xhomalg[105X matrix [3XM[103X. If the underlying ring is
  commutative, the twist is the identity.[133X
  
  [33X[0;0Y(for the installed standard method see [2XEval[102X ([14XC.4-7[114X))[133X
  
  [1X5.5-7 TransposedMatrix[101X
  
  [33X[1;0Y[29X[2XTransposedMatrix[102X( [3XM[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YThe transpose of the [5Xhomalg[105X matrix [3XM[103X.[133X
  
  [33X[0;0Y(for the installed standard method see [2XEval[102X ([14XC.4-8[114X))[133X
  
  [1X5.5-8 CertainRows[101X
  
  [33X[1;0Y[29X[2XCertainRows[102X( [3XM[103X, [3Xplist[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YThe  matrix  of  which  the [22Xi[122X-th row is the [22Xk[122X-th row of the [5Xhomalg[105X matrix [3XM[103X,
  where [22Xk=[122X[3Xplist[103X[22X[i][122X.[133X
  
  [33X[0;0Y(for the installed standard method see [2XEval[102X ([14XC.4-10[114X))[133X
  
  [1X5.5-9 CertainColumns[101X
  
  [33X[1;0Y[29X[2XCertainColumns[102X( [3XM[103X, [3Xplist[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YThe  matrix of which the [22Xj[122X-th column is the [22Xl[122X-th column of the [5Xhomalg[105X matrix
  [3XM[103X, where [22Xl=[122X[3Xplist[103X[22X[j][122X.[133X
  
  [33X[0;0Y(for the installed standard method see [2XEval[102X ([14XC.4-11[114X))[133X
  
  [1X5.5-10 UnionOfRows[101X
  
  [33X[1;0Y[29X[2XUnionOfRows[102X( [[3XR[103X, [3Xnr_cols[103X, ][3XL[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YStack  the  [5Xhomalg[105X matrices in the list [3XL[103X. The entries of [3XL[103X must be matrices
  over  the  [5Xhomalg[105X  ring  [3XR[103X  with  [3Xnr_cols[103X  columns. If [3XL[103X is non-empty, [3XR[103X and
  [3Xnr_cols[103X can be omitted.[133X
  
  [33X[0;0Y(for the installed standard method see [2XEval[102X ([14XC.4-12[114X))[133X
  
  [1X5.5-11 UnionOfColumns[101X
  
  [33X[1;0Y[29X[2XUnionOfColumns[102X( [[3XR[103X, [3Xnr_rows[103X, ][3XL[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YAugment the [5Xhomalg[105X matrices in the list [3XL[103X. The entries of [3XL[103X must be matrices
  over  the  [5Xhomalg[105X ring [3XR[103X with [3Xnr_rows[103X rows. If [3XL[103X is non-empty, [3XR[103X and [3Xnr_rows[103X
  can be omitted.[133X
  
  [33X[0;0Y(for the installed standard method see [2XEval[102X ([14XC.4-13[114X))[133X
  
  [1X5.5-12 ConvertRowToMatrix[101X
  
  [33X[1;0Y[29X[2XConvertRowToMatrix[102X( [3XM[103X, [3Xr[103X, [3Xc[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YFold the row [3XM[103X to an [3Xr[103Xx[3Xc[103X-matrix.[133X
  
  [1X5.5-13 ConvertColumnToMatrix[101X
  
  [33X[1;0Y[29X[2XConvertColumnToMatrix[102X( [3XM[103X, [3Xr[103X, [3Xc[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YFold the column [3XM[103X to an [3Xr[103Xx[3Xc[103X-matrix.[133X
  
  [1X5.5-14 ConvertMatrixToRow[101X
  
  [33X[1;0Y[29X[2XConvertMatrixToRow[102X( [3XM[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YUnfold the matrix [3XM[103X row-wise into a row.[133X
  
  [1X5.5-15 ConvertMatrixToColumn[101X
  
  [33X[1;0Y[29X[2XConvertMatrixToColumn[102X( [3XM[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YUnfold the matrix [3XM[103X column-wise into a column.[133X
  
  [1X5.5-16 DiagMat[101X
  
  [33X[1;0Y[29X[2XDiagMat[102X( [[3XR[103X, ][3Xlist[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YBuild  the  block diagonal matrix out of the [5Xhomalg[105X matrices listed in [3Xlist[103X.
  If [3Xlist[103X is non-empty, [3XR[103X can be omitted.[133X
  
  [33X[0;0Y(for the installed standard method see [2XEval[102X ([14XC.4-14[114X))[133X
  
  [1X5.5-17 KroneckerMat[101X
  
  [33X[1;0Y[29X[2XKroneckerMat[102X( [3XA[103X, [3XB[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YThe Kronecker (or tensor) product of the two [5Xhomalg[105X matrices [3XA[103X and [3XB[103X.[133X
  
  [33X[0;0Y(for the installed standard method see [2XEval[102X ([14XC.4-15[114X))[133X
  
  [1X5.5-18 DualKroneckerMat[101X
  
  [33X[1;0Y[29X[2XDualKroneckerMat[102X( [3XA[103X, [3XB[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YThe dual Kronecker product of the two [5Xhomalg[105X matrices [3XA[103X and [3XB[103X.[133X
  
  [33X[0;0Y(for the installed standard method see [2XEval[102X ([14XC.4-16[114X))[133X
  
  [1X5.5-19 \*[101X
  
  [33X[1;0Y[29X[2X\*[102X( [3Xa[103X, [3XA[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YThe product of the ring element [3Xa[103X with the [5Xhomalg[105X matrix [3XA[103X (enter: [3Xa[103X [10X*[110X [3XA[103X;).[133X
  
  [33X[0;0Y(for the installed standard method see [2XEval[102X ([14XC.4-17[114X))[133X
  
  [1X5.5-20 \+[101X
  
  [33X[1;0Y[29X[2X\+[102X( [3XA[103X, [3XB[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YThe sum of the two [5Xhomalg[105X matrices [3XA[103X and [3XB[103X (enter: [3XA[103X [10X+[110X [3XB[103X;).[133X
  
  [33X[0;0Y(for the installed standard method see [2XEval[102X ([14XC.4-18[114X))[133X
  
  [1X5.5-21 \-[101X
  
  [33X[1;0Y[29X[2X\-[102X( [3XA[103X, [3XB[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YThe difference of the two [5Xhomalg[105X matrices [3XA[103X and [3XB[103X (enter: [3XA[103X [10X-[110X [3XB[103X;).[133X
  
  [33X[0;0Y(for the installed standard method see [2XEval[102X ([14XC.4-19[114X))[133X
  
  [1X5.5-22 \*[101X
  
  [33X[1;0Y[29X[2X\*[102X( [3XA[103X, [3XB[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YThe matrix product of the two [5Xhomalg[105X matrices [3XA[103X and [3XB[103X (enter: [3XA[103X [10X*[110X [3XB[103X;).[133X
  
  [33X[0;0Y(for the installed standard method see [2XEval[102X ([14XC.4-20[114X))[133X
  
  [1X5.5-23 \=[101X
  
  [33X[1;0Y[29X[2X\=[102X( [3XA[103X, [3XB[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck  if  the  [5Xhomalg[105X  matrices  [3XA[103X  and [3XB[103X are equal (enter: [3XA[103X [10X=[110X [3XB[103X;), taking
  possible ring relations into account.[133X
  
  [33X[0;0Y(for the installed standard method see [2XAreEqualMatrices[102X ([14XB.2-1[114X))[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZZ := HomalgRingOfIntegers( );[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27XA := HomalgMatrix( "[ 1 ]", ZZ );[127X[104X
    [4X[28X<A 1 x 1 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XB := HomalgMatrix( "[ 3 ]", ZZ );[127X[104X
    [4X[28X<A 1 x 1 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XZ2 := ZZ / 2;[127X[104X
    [4X[28XZ/( 2 )[128X[104X
    [4X[25Xgap>[125X [27XA := Z2 * A;[127X[104X
    [4X[28X<A 1 x 1 matrix over a residue class ring>[128X[104X
    [4X[25Xgap>[125X [27XB := Z2 * B;[127X[104X
    [4X[28X<A 1 x 1 matrix over a residue class ring>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( A );[127X[104X
    [4X[28X[ [  1 ] ][128X[104X
    [4X[28X[128X[104X
    [4X[28Xmodulo [ 2 ][128X[104X
    [4X[25Xgap>[125X [27XDisplay( B );[127X[104X
    [4X[28X[ [  3 ] ][128X[104X
    [4X[28X[128X[104X
    [4X[28Xmodulo [ 2 ][128X[104X
    [4X[25Xgap>[125X [27XA = B;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X5.5-24 GetColumnIndependentUnitPositions[101X
  
  [33X[1;0Y[29X[2XGetColumnIndependentUnitPositions[102X( [3XA[103X, [3Xposlist[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya (possibly empty) list of pairs of positive integers[133X
  
  [33X[0;0YThe  list  of column independet unit position of the matrix [3XA[103X. We say that a
  unit  [3XA[103X[22X[i,k][122X  is column independet from the unit [3XA[103X[22X[l,j][122X if [22Xi>l[122X and [3XA[103X[22X[l,k]=0[122X.
  The  rows are scanned from top to bottom and within each row the columns are
  scanned  from right to left searching for new units, column independent from
  the preceding ones. If [3XA[103X[22X[i,k][122X is a new column independent unit then [22X[i,k][122X is
  added to the output list. If [3XA[103X has no units the empty list is returned.[133X
  
  [33X[0;0Y(for  the  installed  standard  method see [2XGetColumnIndependentUnitPositions[102X
  ([14XB.2-6[114X))[133X
  
  [1X5.5-25 GetRowIndependentUnitPositions[101X
  
  [33X[1;0Y[29X[2XGetRowIndependentUnitPositions[102X( [3XA[103X, [3Xposlist[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya (possibly empty) list of pairs of positive integers[133X
  
  [33X[0;0YThe list of row independet unit position of the matrix [3XA[103X. We say that a unit
  [3XA[103X[22X[k,j][122X  is  row  independet  from  the  unit [3XA[103X[22X[i,l][122X if [22Xj>l[122X and [3XA[103X[22X[k,l]=0[122X. The
  columns  are  scanned from left to right and within each column the rows are
  scanned from bottom to top searching for new units, row independent from the
  preceding  ones.  If  [3XA[103X[22X[k,j][122X  is  a new row independent unit then [22X[j,k][122X (yes
  [22X[j,k][122X)  is  added  to  the  output list. If [3XA[103X has no units the empty list is
  returned.[133X
  
  [33X[0;0Y(for   the  installed  standard  method  see  [2XGetRowIndependentUnitPositions[102X
  ([14XB.2-7[114X))[133X
  
  [1X5.5-26 GetUnitPosition[101X
  
  [33X[1;0Y[29X[2XGetUnitPosition[102X( [3XA[103X, [3Xposlist[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya (possibly empty) list of pairs of positive integers[133X
  
  [33X[0;0YThe  position [22X[i,j][122X of the first unit [3XA[103X[22X[i,j][122X in the matrix [3XA[103X, where the rows
  are  scanned  from top to bottom and within each row the columns are scanned
  from  left  to  right.  If [3XA[103X[22X[i,j][122X is the first occurrence of a unit then the
  position pair [22X[i,j][122X is returned. Otherwise [10Xfail[110X is returned.[133X
  
  [33X[0;0Y(for the installed standard method see [2XGetUnitPosition[102X ([14XB.2-8[114X))[133X
  
  [1X5.5-27 Eliminate[101X
  
  [33X[1;0Y[29X[2XEliminate[102X( [3Xrel[103X, [3Xindets[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YEliminate the independents [3Xindets[103X from the matrix (or list of ring elements)
  [3Xrel[103X,  i.e. compute a generating set of the ideal defined as the intersection
  of  the  ideal  generated  by  the  entries of the list [3Xrel[103X with the subring
  generated  by  all  indeterminates  except  those  in [3Xindets[103X. by the list of
  indeterminates [3Xindets[103X.[133X
  
  [1X5.5-28 BasisOfRowModule[101X
  
  [33X[1;0Y[29X[2XBasisOfRowModule[102X( [3XM[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YLet  [22XR[122X be the ring over which [3XM[103X is defined ([22XR:=[122X[10XHomalgRing[110X( [3XM[103X )) and [22XS[122X be the
  row  span  of  [3XM[103X,  i.e.  the  [22XR[122X-submodule  of  the  free  left module [22XR^(1 ×
  NrColumns(  [3XM[103X  ))[122X  spanned  by  the  rows  of [3XM[103X. A solution to the [21Xsubmodule
  membership  problem[121X is an algorithm which can decide if an element [22Xm[122X in [22XR^(1
  ×  NrColumns(  [3XM[103X  ))[122X is contained in [22XS[122X or not. And exactly like the Gaussian
  (resp.  Hermite) normal form when [22XR[122X is a field (resp. principal ideal ring),
  the  row  span of the resulting matrix [22XB[122X coincides with the row span [22XS[122X of [3XM[103X,
  and  computing  [22XB[122X  is  typically  the  first step of such an algorithm. (-->
  Appendix [14XA[114X)[133X
  
  [1X5.5-29 BasisOfColumnModule[101X
  
  [33X[1;0Y[29X[2XBasisOfColumnModule[102X( [3XM[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YLet  [22XR[122X be the ring over which [3XM[103X is defined ([22XR:=[122X[10XHomalgRing[110X( [3XM[103X )) and [22XS[122X be the
  column span of [3XM[103X, i.e. the [22XR[122X-submodule of the free right module [22XR^(NrRows( [3XM[103X
  )  ×  1)[122X spanned by the columns of [3XM[103X. A solution to the [21Xsubmodule membership
  problem[121X is an algorithm which can decide if an element [22Xm[122X in [22XR^(NrRows( [3XM[103X ) ×
  1)[122X  is  contained in [22XS[122X or not. And exactly like the Gaussian (resp. Hermite)
  normal  form when [22XR[122X is a field (resp. principal ideal ring), the column span
  of  the  resulting  matrix  [22XB[122X  coincides  with  the  column span [22XS[122X of [3XM[103X, and
  computing  [22XB[122X is typically the first step of such an algorithm. (--> Appendix
  [14XA[114X)[133X
  
  [1X5.5-30 DecideZeroRows[101X
  
  [33X[1;0Y[29X[2XDecideZeroRows[102X( [3XA[103X, [3XB[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YLet  [3XA[103X  and [3XB[103X be matrices having the same number of columns and defined over
  the  same  ring  [22XR[122X  ([22X:=[122X[10XHomalgRing[110X( [3XA[103X )) and [22XS[122X be the row span of [3XB[103X, i.e. the
  [22XR[122X-submodule  of  the  free left module [22XR^(1 × NrColumns( [3XB[103X ))[122X spanned by the
  rows  of  [3XB[103X.  The result is a matrix [22XC[122X having the same shape as [3XA[103X, for which
  the  [22Xi[122X-th  row  [22X[3XC[103X^i[122X  is  equivalent  to the [22Xi[122X-th row [22X[3XA[103X^i[122X of [3XA[103X modulo [22XS[122X, i.e.
  [22X[3XC[103X^i-[3XA[103X^i[122X is an element of the row span [22XS[122X of [3XB[103X. Moreover, the row [22X[3XC[103X^i[122X is zero,
  if  and  only  if  the row [22X[3XA[103X^i[122X is an element of [22XS[122X. So [10XDecideZeroRows[110X decides
  which rows of [3XA[103X are zero modulo the rows of [3XB[103X. (--> Appendix [14XA[114X)[133X
  
  [1X5.5-31 DecideZeroColumns[101X
  
  [33X[1;0Y[29X[2XDecideZeroColumns[102X( [3XA[103X, [3XB[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YLet  [3XA[103X and [3XB[103X be matrices having the same number of rows and defined over the
  same  ring  [22XR[122X  ([22X:=[122X[10XHomalgRing[110X(  [3XA[103X  )) and [22XS[122X be the column span of [3XB[103X, i.e. the
  [22XR[122X-submodule  of  the  free  right  module [22XR^(NrRows( [3XB[103X ) × 1)[122X spanned by the
  columns of [3XB[103X. The result is a matrix [22XC[122X having the same shape as [3XA[103X, for which
  the [22Xi[122X-th column [22X[3XC[103X_i[122X is equivalent to the [22Xi[122X-th column [22X[3XA[103X_i[122X of [3XA[103X modulo [22XS[122X, i.e.
  [22X[3XC[103X_i-[3XA[103X_i[122X is an element of the column span [22XS[122X of [3XB[103X. Moreover, the column [22X[3XC[103X_i[122X is
  zero, if and only if the column [22X[3XA[103X_i[122X is an element of [22XS[122X. So [10XDecideZeroColumns[110X
  decides  which  columns of [3XA[103X are zero modulo the columns of [3XB[103X. (--> Appendix
  [14XA[114X)[133X
  
  [1X5.5-32 SyzygiesGeneratorsOfRows[101X
  
  [33X[1;0Y[29X[2XSyzygiesGeneratorsOfRows[102X( [3XM[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YLet  [22XR[122X  be the ring over which [3XM[103X is defined ([22XR:=[122X[10XHomalgRing[110X( [3XM[103X )). The matrix
  of  row  syzygies  [10XSyzygiesGeneratorsOfRows[110X( [3XM[103X ) is a matrix whose rows span
  the  left  kernel  of [3XM[103X, i.e. the [22XR[122X-submodule of the free left module [22XR^(1 ×
  NrRows( [3XM[103X ))[122X consisting of all rows [22XX[122X satisfying [22XX[3XM[103X=0[122X. (--> Appendix [14XA[114X)[133X
  
  [1X5.5-33 SyzygiesGeneratorsOfColumns[101X
  
  [33X[1;0Y[29X[2XSyzygiesGeneratorsOfColumns[102X( [3XM[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YLet  [22XR[122X  be the ring over which [3XM[103X is defined ([22XR:=[122X[10XHomalgRing[110X( [3XM[103X )). The matrix
  of  column  syzygies  [10XSyzygiesGeneratorsOfColumns[110X(  [3XM[103X  )  is  a matrix whose
  columns  span  the right kernel of [3XM[103X, i.e. the [22XR[122X-submodule of the free right
  module  [22XR^(NrColumns(  [3XM[103X ) × 1)[122X consisting of all columns [22XX[122X satisfying [22X[3XM[103XX=0[122X.
  (--> Appendix [14XA[114X)[133X
  
  [1X5.5-34 SyzygiesGeneratorsOfRows[101X
  
  [33X[1;0Y[29X[2XSyzygiesGeneratorsOfRows[102X( [3XM[103X, [3XM2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YLet  [22XR[122X  be the ring over which [3XM[103X is defined ([22XR:=[122X[10XHomalgRing[110X( [3XM[103X )). The matrix
  of [13Xrelative[113X row syzygies [10XSyzygiesGeneratorsOfRows[110X( [3XM[103X, [3XM2[103X ) is a matrix whose
  rows  span  the left kernel of [3XM[103X modulo [3XM2[103X, i.e. the [22XR[122X-submodule of the free
  left module [22XR^(1 × NrRows( [3XM[103X ))[122X consisting of all rows [22XX[122X satisfying [22XX[3XM[103X+Y[3XM2[103X=0[122X
  for some row [22XY ∈ R^(1 × NrRows( [3XM2[103X ))[122X. (--> Appendix [14XA[114X)[133X
  
  [1X5.5-35 SyzygiesGeneratorsOfColumns[101X
  
  [33X[1;0Y[29X[2XSyzygiesGeneratorsOfColumns[102X( [3XM[103X, [3XM2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YLet  [22XR[122X  be the ring over which [3XM[103X is defined ([22XR:=[122X[10XHomalgRing[110X( [3XM[103X )). The matrix
  of [13Xrelative[113X column syzygies [10XSyzygiesGeneratorsOfColumns[110X( [3XM[103X, [3XM2[103X ) is a matrix
  whose  columns span the right kernel of [3XM[103X modulo [3XM2[103X, i.e. the [22XR[122X-submodule of
  the  free  right  module  [22XR^(NrColumns( [3XM[103X ) × 1)[122X consisting of all columns [22XX[122X
  satisfying  [22X[3XM[103XX+[3XM2[103XY=0[122X  for  some  column  [22XY  ∈  R^(NrColumns( [3XM2[103X ) × 1)[122X. (-->
  Appendix [14XA[114X)[133X
  
  [1X5.5-36 ReducedBasisOfRowModule[101X
  
  [33X[1;0Y[29X[2XReducedBasisOfRowModule[102X( [3XM[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YLike  [10XBasisOfRowModule[110X(  [3XM[103X  ) but where the matrix [10XSyzygiesGeneratorsOfRows[110X(
  [10XReducedBasisOfRowModule[110X(  [3XM[103X  )  )  contains  no  units.  This  can easily be
  achieved    starting    from    [22XB:=[122X[10XBasisOfRowModule[110X(    [3XM[103X   )   (and   using
  [2XGetColumnIndependentUnitPositions[102X  ([14X5.5-24[114X)  applied  to  the  matrix of row
  syzygies of [22XB[122X, etc). (--> Appendix [14XA[114X)[133X
  
  [1X5.5-37 ReducedBasisOfColumnModule[101X
  
  [33X[1;0Y[29X[2XReducedBasisOfColumnModule[102X( [3XM[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YLike     [10XBasisOfColumnModule[110X(     [3XM[103X     )     but     where    the    matrix
  [10XSyzygiesGeneratorsOfColumns[110X(  [10XReducedBasisOfColumnModule[110X(  [3XM[103X ) ) contains no
  units. This can easily be achieved starting from [22XB:=[122X[10XBasisOfColumnModule[110X( [3XM[103X )
  (and  using [2XGetRowIndependentUnitPositions[102X ([14X5.5-25[114X) applied to the matrix of
  column syzygies of [22XB[122X, etc.). (--> Appendix [14XA[114X)[133X
  
  [1X5.5-38 ReducedSyzygiesGeneratorsOfRows[101X
  
  [33X[1;0Y[29X[2XReducedSyzygiesGeneratorsOfRows[102X( [3XM[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YLike    [10XSyzygiesGeneratorsOfRows[110X(    [3XM[103X    )    but    where    the    matrix
  [10XSyzygiesGeneratorsOfRows[110X( [10XReducedSyzygiesGeneratorsOfRows[110X( [3XM[103X ) ) contains no
  units.     This     can     easily     be     achieved     starting     from
  [22XC:=[122X[10XSyzygiesGeneratorsOfRows[110X(          [3XM[103X         )         (and         using
  [2XGetColumnIndependentUnitPositions[102X  ([14X5.5-24[114X)  applied  to  the  matrix of row
  syzygies of [22XC[122X, etc.). (--> Appendix [14XA[114X)[133X
  
  [1X5.5-39 ReducedSyzygiesGeneratorsOfColumns[101X
  
  [33X[1;0Y[29X[2XReducedSyzygiesGeneratorsOfColumns[102X( [3XM[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YLike    [10XSyzygiesGeneratorsOfColumns[110X(    [3XM[103X    )    but   where   the   matrix
  [10XSyzygiesGeneratorsOfColumns[110X(   [10XReducedSyzygiesGeneratorsOfColumns[110X(   [3XM[103X  )  )
  contains   no   units.   This   can   easily   be   achieved  starting  from
  [22XC:=[122X[10XSyzygiesGeneratorsOfColumns[110X(         [3XM[103X         )        (and        using
  [2XGetRowIndependentUnitPositions[102X  ([14X5.5-25[114X)  applied  to  the  matrix of column
  syzygies of [22XC[122X, etc.). (--> Appendix [14XA[114X)[133X
  
  [1X5.5-40 BasisOfRowsCoeff[101X
  
  [33X[1;0Y[29X[2XBasisOfRowsCoeff[102X( [3XM[103X, [3XT[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YReturns  [22XB:=[122X[10XBasisOfRowModule[110X(  [3XM[103X  )  and  assigns  the  [13Xvoid[113X  matrix  [3XT[103X (-->
  [2XHomalgVoidMatrix[102X ([14X5.2-5[114X)) such that [22XB = [3XT[103X [3XM[103X[122X. (--> Appendix [14XA[114X)[133X
  
  [1X5.5-41 BasisOfColumnsCoeff[101X
  
  [33X[1;0Y[29X[2XBasisOfColumnsCoeff[102X( [3XM[103X, [3XT[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YReturns  [22XB:=[122X[10XBasisOfRowModule[110X(  [3XM[103X  )  and  assigns  the  [13Xvoid[113X  matrix  [3XT[103X (-->
  [2XHomalgVoidMatrix[102X ([14X5.2-5[114X)) such that [22XB = [3XM[103X [3XT[103X[122X. (--> Appendix [14XA[114X)[133X
  
  [1X5.5-42 DecideZeroRowsEffectively[101X
  
  [33X[1;0Y[29X[2XDecideZeroRowsEffectively[102X( [3XA[103X, [3XB[103X, [3XT[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YReturns  [22XM:=[122X[10XDecideZeroRows[110X(  [3XA[103X,  [3XB[103X  )  and  assigns  the  [13Xvoid[113X matrix [3XT[103X (-->
  [2XHomalgVoidMatrix[102X ([14X5.2-5[114X)) such that [22XM = [3XA[103X + [3XT[103X[3XB[103X[122X. (--> Appendix [14XA[114X)[133X
  
  [1X5.5-43 DecideZeroColumnsEffectively[101X
  
  [33X[1;0Y[29X[2XDecideZeroColumnsEffectively[102X( [3XA[103X, [3XB[103X, [3XT[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YReturns  [22XM:=[122X[10XDecideZeroColumns[110X(  [3XA[103X,  [3XB[103X  )  and assigns the [13Xvoid[113X matrix [3XT[103X (-->
  [2XHomalgVoidMatrix[102X ([14X5.2-5[114X)) such that [22XM = [3XA[103X + [3XB[103X[3XT[103X[122X. (--> Appendix [14XA[114X)[133X
  
  [1X5.5-44 BasisOfRows[101X
  
  [33X[1;0Y[29X[2XBasisOfRows[102X( [3XM[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XBasisOfRows[102X( [3XM[103X, [3XT[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YWith  one  argument  it  is a synonym of [2XBasisOfRowModule[102X ([14X5.5-28[114X). with two
  arguments it is a synonym of [2XBasisOfRowsCoeff[102X ([14X5.5-40[114X).[133X
  
  [1X5.5-45 BasisOfColumns[101X
  
  [33X[1;0Y[29X[2XBasisOfColumns[102X( [3XM[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XBasisOfColumns[102X( [3XM[103X, [3XT[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YWith  one argument it is a synonym of [2XBasisOfColumnModule[102X ([14X5.5-29[114X). with two
  arguments it is a synonym of [2XBasisOfColumnsCoeff[102X ([14X5.5-41[114X).[133X
  
  [1X5.5-46 DecideZero[101X
  
  [33X[1;0Y[29X[2XDecideZero[102X( [3Xmat[103X, [3Xrel[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [4X[32X  Code  [32X[104X
    [4XInstallMethod( DecideZero,[104X
    [4X        "for sets of ring relations",[104X
    [4X        [ IsHomalgMatrix, IsHomalgRingRelations ],[104X
    [4X        [104X
    [4X  function( mat, rel )[104X
    [4X    [104X
    [4X    return DecideZero( mat,  MatrixOfRelations( rel ) );[104X
    [4X    [104X
    [4Xend );[104X
  [4X[32X[104X
  
  [1X5.5-47 SyzygiesOfRows[101X
  
  [33X[1;0Y[29X[2XSyzygiesOfRows[102X( [3XM[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSyzygiesOfRows[102X( [3XM[103X, [3XM2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YWith one argument it is a synonym of [2XSyzygiesGeneratorsOfRows[102X ([14X5.5-32[114X). with
  two arguments it is a synonym of [2XSyzygiesGeneratorsOfRows[102X ([14X5.5-34[114X).[133X
  
  [1X5.5-48 SyzygiesOfColumns[101X
  
  [33X[1;0Y[29X[2XSyzygiesOfColumns[102X( [3XM[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSyzygiesOfColumns[102X( [3XM[103X, [3XM2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YWith  one  argument it is a synonym of [2XSyzygiesGeneratorsOfColumns[102X ([14X5.5-33[114X).
  with two arguments it is a synonym of [2XSyzygiesGeneratorsOfColumns[102X ([14X5.5-35[114X).[133X
  
  [1X5.5-49 ReducedSyzygiesOfRows[101X
  
  [33X[1;0Y[29X[2XReducedSyzygiesOfRows[102X( [3XM[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XReducedSyzygiesOfRows[102X( [3XM[103X, [3XM2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YWith  one  argument  it  is  a  synonym  of  [2XReducedSyzygiesGeneratorsOfRows[102X
  ([14X5.5-38[114X).    With    two   arguments   it   calls   [10XReducedBasisOfRowModule[110X(
  [10XSyzygiesGeneratorsOfRows[110X(  [3XM[103X,  [3XM2[103X ) ). (--> [2XReducedBasisOfRowModule[102X ([14X5.5-36[114X)
  and [2XSyzygiesGeneratorsOfRows[102X ([14X5.5-34[114X))[133X
  
  [1X5.5-50 ReducedSyzygiesOfColumns[101X
  
  [33X[1;0Y[29X[2XReducedSyzygiesOfColumns[102X( [3XM[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XReducedSyzygiesOfColumns[102X( [3XM[103X, [3XM2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YWith  one  argument  it  is  a synonym of [2XReducedSyzygiesGeneratorsOfColumns[102X
  ([14X5.5-39[114X).   With   two   arguments   it   calls  [10XReducedBasisOfColumnModule[110X(
  [10XSyzygiesGeneratorsOfColumns[110X(  [3XM[103X,  [3XM2[103X  )  ).  (--> [2XReducedBasisOfColumnModule[102X
  ([14X5.5-37[114X) and [2XSyzygiesGeneratorsOfColumns[102X ([14X5.5-35[114X))[133X
  
  [1X5.5-51 RightDivide[101X
  
  [33X[1;0Y[29X[2XRightDivide[102X( [3XB[103X, [3XA[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix or fail[133X
  
  [33X[0;0YLet  [3XB[103X  and [3XA[103X be matrices having the same number of columns and defined over
  the  same  ring.  The matrix [10XRightDivide[110X( [3XB[103X, [3XA[103X ) is a particular solution of
  the  inhomogeneous (one sided) linear system of equations [22XX[3XA[103X=[3XB[103X[122X in case it is
  solvable. Otherwise [10Xfail[110X is returned. The name [10XRightDivide[110X suggests [21X[22XX=[3XB[103X[3XA[103X^-1[122X[121X.
  This  generalizes  [2XLeftInverse[102X  ([14X5.5-2[114X)  for  which  [3XB[103X  becomes the identity
  matrix. (--> [2XSyzygiesGeneratorsOfRows[102X ([14X5.5-32[114X))[133X
  
  [1X5.5-52 LeftDivide[101X
  
  [33X[1;0Y[29X[2XLeftDivide[102X( [3XA[103X, [3XB[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix or fail[133X
  
  [33X[0;0YLet  [3XA[103X and [3XB[103X be matrices having the same number of rows and defined over the
  same  ring.  The  matrix  [10XLeftDivide[110X( [3XA[103X, [3XB[103X ) is a particular solution of the
  inhomogeneous  (one  sided)  linear  system  of equations [22X[3XA[103XX=[3XB[103X[122X in case it is
  solvable.  Otherwise [10Xfail[110X is returned. The name [10XLeftDivide[110X suggests [21X[22XX=[3XA[103X^-1[3XB[103X[122X[121X.
  This  generalizes  [2XRightInverse[102X  ([14X5.5-3[114X)  for  which  [3XB[103X becomes the identity
  matrix. (--> [2XSyzygiesGeneratorsOfColumns[102X ([14X5.5-33[114X))[133X
  
  [1X5.5-53 RightDivide[101X
  
  [33X[1;0Y[29X[2XRightDivide[102X( [3XB[103X, [3XA[103X, [3XL[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix or fail[133X
  
  [33X[0;0YLet  [3XB[103X,  [3XA[103X  and  [3XL[103X be matrices having the same number of columns and defined
  over  the  same  ring.  The  matrix  [10XRightDivide[110X(  [3XB[103X, [3XA[103X, [3XL[103X ) is a particular
  solution of the inhomogeneous (one sided) linear system of equations [22XX[3XA[103X+Y[3XL[103X=[3XB[103X[122X
  in  case  it  is solvable (for some [22XY[122X which is forgotten). Otherwise [10Xfail[110X is
  returned.  The  name  [10XRightDivide[110X  suggests  [21X[22XX=[3XB[103X[3XA[103X^-1[122X  modulo  [3XL[103X[121X. (Cf. [BR08,
  Subsection 3.1.1])[133X
  
  [4X[32X  Code  [32X[104X
    [4XInstallMethod( RightDivide,[104X
    [4X        "for homalg matrices",[104X
    [4X        [ IsHomalgMatrix, IsHomalgMatrix, IsHomalgMatrix ],[104X
    [4X        [104X
    [4X  function( B, A, L ) ## CAUTION: Do not use lazy evaluation here!!![104X
    [4X    local R, BL, ZA, AL, ZB, T, B_;[104X
    [4X    [104X
    [4X    R := HomalgRing( B );[104X
    [4X    [104X
    [4X    BL := BasisOfRows( L );[104X
    [4X    [104X
    [4X    ## first reduce A modulo L[104X
    [4X    ZA := DecideZeroRows( A, BL );[104X
    [4X    [104X
    [4X    AL := UnionOfRows( ZA, BL );[104X
    [4X    [104X
    [4X    ## also reduce B modulo L[104X
    [4X    ZB := DecideZeroRows( B, BL );[104X
    [4X    [104X
    [4X    ## B_ = ZB + T * AL[104X
    [4X    T := HomalgVoidMatrix( R );[104X
    [4X    B_ := DecideZeroRowsEffectively( ZB, AL, T );[104X
    [4X    [104X
    [4X    ## if B_ does not vanish[104X
    [4X    if not IsZero( B_ ) then[104X
    [4X        return fail;[104X
    [4X    fi;[104X
    [4X    [104X
    [4X    T := CertainColumns( T, [ 1 .. NrRows( A ) ] );[104X
    [4X    [104X
    [4X    ## check assertion[104X
    [4X    Assert( 5, IsZero( DecideZeroRows( B + T * A, BL ) ) );[104X
    [4X    [104X
    [4X    return -T;[104X
    [4X    [104X
    [4Xend );[104X
  [4X[32X[104X
  
  [1X5.5-54 LeftDivide[101X
  
  [33X[1;0Y[29X[2XLeftDivide[102X( [3XA[103X, [3XB[103X, [3XL[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix or fail[133X
  
  [33X[0;0YLet  [3XA[103X,  [3XB[103X  and  [3XL[103X be matrices having the same number of columns and defined
  over  the  same  ring.  The  matrix  [10XLeftDivide[110X(  [3XA[103X,  [3XB[103X, [3XL[103X ) is a particular
  solution of the inhomogeneous (one sided) linear system of equations [22X[3XA[103XX+[3XL[103XY=[3XB[103X[122X
  in  case  it  is solvable (for some [22XY[122X which is forgotten). Otherwise [10Xfail[110X is
  returned.  The  name  [10XLeftDivide[110X  suggests  [21X[22XX=[3XA[103X^-1[3XB[103X[122X  modulo  [3XL[103X[121X.  (Cf. [BR08,
  Subsection 3.1.1])[133X
  
  [4X[32X  Code  [32X[104X
    [4XInstallMethod( LeftDivide,[104X
    [4X        "for homalg matrices",[104X
    [4X        [ IsHomalgMatrix, IsHomalgMatrix, IsHomalgMatrix ],[104X
    [4X        [104X
    [4X  function( A, B, L ) ## CAUTION: Do not use lazy evaluation here!!![104X
    [4X    local R, BL, ZA, AL, ZB, T, B_;[104X
    [4X    [104X
    [4X    R := HomalgRing( B );[104X
    [4X    [104X
    [4X    BL := BasisOfColumns( L );[104X
    [4X    [104X
    [4X    ## first reduce A modulo L[104X
    [4X    ZA := DecideZeroColumns( A, BL );[104X
    [4X    [104X
    [4X    AL := UnionOfColumns( ZA, BL );[104X
    [4X    [104X
    [4X    ## also reduce B modulo L[104X
    [4X    ZB := DecideZeroColumns( B, BL );[104X
    [4X    [104X
    [4X    ## B_ = ZB + AL * T[104X
    [4X    T := HomalgVoidMatrix( R );[104X
    [4X    B_ := DecideZeroColumnsEffectively( ZB, AL, T );[104X
    [4X    [104X
    [4X    ## if B_ does not vanish[104X
    [4X    if not IsZero( B_ ) then[104X
    [4X        return fail;[104X
    [4X    fi;[104X
    [4X    [104X
    [4X    T := CertainRows( T, [ 1 .. NrColumns( A ) ] );[104X
    [4X    [104X
    [4X    ## check assertion[104X
    [4X    Assert( 5, IsZero( DecideZeroColumns( B + A * T, BL ) ) );[104X
    [4X    [104X
    [4X    return -T;[104X
    [4X    [104X
    [4Xend );[104X
  [4X[32X[104X
  
  [1X5.5-55 GenerateSameRowModule[101X
  
  [33X[1;0Y[29X[2XGenerateSameRowModule[102X( [3XM[103X, [3XN[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck  if  the  row span of [3XM[103X and of [3XN[103X are identical or not (--> [2XRightDivide[102X
  ([14X5.5-51[114X)).[133X
  
  [1X5.5-56 GenerateSameColumnModule[101X
  
  [33X[1;0Y[29X[2XGenerateSameColumnModule[102X( [3XM[103X, [3XN[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck  if the column span of [3XM[103X and of [3XN[103X are identical or not (--> [2XLeftDivide[102X
  ([14X5.5-52[114X)).[133X
  
  [1X5.5-57 SimplifyHomalgMatrixByLeftAndRightMultiplicationWithInvertibleMatrices[101X
  
  [33X[1;0Y[29X[2XSimplifyHomalgMatrixByLeftAndRightMultiplicationWithInvertibleMatrices[102X( [3XM[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list of 5 [5Xhomalg[105X matrices[133X
  
  [33X[0;0YThe  input  is a [5Xhomalg[105X matrix [3XM[103X. The output is a 5-tuple of [5Xhomalg[105X matrices
  [3XS[103X, [3XU[103X, [3XV[103X, [3XUI[103X, [3XVI[103X, such that [3X U M V = S[103X. Moreover, [3XU[103X and [3XV[103X are invertible with
  inverses  [3XUI[103X,  [3XVI[103X,  respectively.  The idea is that the matrix [3XS[103X should look
  "simpler" than [3XM[103X.[133X
  
  [1X5.5-58 SimplifyHomalgMatrixByLeftMultiplicationWithInvertibleMatrix[101X
  
  [33X[1;0Y[29X[2XSimplifyHomalgMatrixByLeftMultiplicationWithInvertibleMatrix[102X( [3XM[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list of 3 [5Xhomalg[105X matrices[133X
  
  [33X[0;0YThe  input  is a [5Xhomalg[105X matrix [3XM[103X. The output is a 3-tuple of [5Xhomalg[105X matrices
  [3XS[103X,  [3XT[103X, [3XTI[103X such that [3X T M = S[103X. Moreover, [3XT[103X is invertible with inverse [3XTI[103X. The
  idea is that the matrix [3XS[103X should look "simpler" than [3XM[103X.[133X
  
  [1X5.5-59 SimplifyHomalgMatrixByRightMultiplicationWithInvertibleMatrix[101X
  
  [33X[1;0Y[29X[2XSimplifyHomalgMatrixByRightMultiplicationWithInvertibleMatrix[102X( [3XM[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list of 3 [5Xhomalg[105X matrices[133X
  
  [33X[0;0YThe  input  is a [5Xhomalg[105X matrix [3XM[103X. The output is a 3-tuple of [5Xhomalg[105X matrices
  [3XS[103X,  [3XT[103X, [3XTI[103X such that [3X M T = S[103X. Moreover, [3XT[103X is invertible with inverse [3XTI[103X. The
  idea is that the matrix [3XS[103X should look "simpler" than [3XM[103X.[133X
  
  [1X5.5-60 CoefficientsWithGivenMonomials[101X
  
  [33X[1;0Y[29X[2XCoefficientsWithGivenMonomials[102X( [3XM[103X, [3Xmonomials[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X matrix[133X
  
  [33X[0;0YLet  [10XR  :=  HomalgRing([110X[3XM[103X[10X)[110X.  [10Xmonomials[110X  must be a [5Xhomalg[105X matrix with the same
  number  of columns as [10XM[110X consisting of monomials of [10XR[110X. This method computes a
  [5Xhomalg[105X  matrix [10Xcoeffs[110X (with entries in the coefficients ring of [10XR[110X, yet still
  considered  as  elements  of [10XR[110X) such that [10XM = coeffs * monomials[110X. If no such
  matrix  exists, the behavior is undefined. If the first argument is a [5Xhomalg[105X
  ring  element,  it  is viewed as a [5Xhomalg[105X matrix with a single entry. If the
  second argument is a list of monomials, it is viewed as a column matrix with
  the list elements as entries.[133X
  
