chapA.txt

  
  A The Matrix Tool Operations
  
  The  functions  listed below are components of the homalgTable object stored
  in  the  ring.  They are only indirectly accessible through standard methods
  that invoke them.
  
  
  A.1 The Tool Operations without a Fallback Method
  
  There  are  matrix  methods  for  which homalg needs a homalgTable entry for
  non-internal  rings,  as it cannot provide a suitable fallback. Below is the
  list of these homalgTable entries.
  
  
  A.2 The Tool Operations with a Fallback Method
  
  These are the methods for which it is recommended for performance reasons to
  have  a  homalgTable  entry  for  non-internal rings. homalg only provides a
  generic fallback method.
  
  A.2-1 MonomialMatrix
  
  MonomialMatrix( d, R )  operation
  Returns:  a homalg matrix
  
  The column matrix of d-th monomials of the homalg graded ring R.
  
    Example  
    gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;
    gap> S := GradedRing( R );;
    gap> m := MonomialMatrix( 2, S );
    <A ? x 1 matrix over a graded ring>
    gap> NrRows( m );
    6
    gap> m;
    <A 6 x 1 matrix over a graded ring>
    gap> Display( m );
    x^2,
    x*y,
    x*z,
    y^2,
    y*z,
    z^2
    (over a graded ring) 
  
  
  A.2-2 RandomMatrixBetweenGradedFreeLeftModules
  
  RandomMatrixBetweenGradedFreeLeftModules( degreesS, degreesT, R )  operation
  Returns:  a homalg matrix
  
  A  random r × c-matrix between the graded free left modules R^(-degreesS) ->
  R^(-degreesT), where r =Length(degreesS) and c =Length(degreesT).
  
    Example  
    gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c";;
    gap> S := GradedRing( R );;
    gap> rand := RandomMatrixBetweenGradedFreeLeftModules( [ 2, 3, 4 ], [ 1, 2 ], S );
    <A 3 x 2 matrix over a graded ring>
    gap> #Display( rand );
    gap> #a-2*b+2*c,                                                2,                 
    gap> #a^2-a*b+b^2-2*b*c+5*c^2,                                  3*c,               
    gap> #2*a^3-3*a^2*b+2*a*b^2+3*a^2*c+a*b*c-2*b^2*c-3*b*c^2-2*c^3,a^2-4*a*b-3*a*c-c^2
  
  
  A.2-3 RandomMatrixBetweenGradedFreeRightModules
  
  RandomMatrixBetweenGradedFreeRightModules( degreesT, degreesS, R )  operation
  Returns:  a homalg matrix
  
  A random r × c-matrix between the graded free right modules R^(-degreesS) ->
  R^(-degreesT), where r =Length(degreesT) and c =Length(degreesS).
  
    Example  
    gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c";;
    gap> S := GradedRing( R );;
    gap> rand := RandomMatrixBetweenGradedFreeRightModules( [ 1, 2 ], [ 2, 3, 4 ], S );
    <A 2 x 3 matrix over a graded ring>
    gap> #Display( rand );
    gap> #a-2*b-c,a*b+b^2-b*c,2*a^3-a*b^2-4*b^3+4*a^2*c-3*a*b*c-b^2*c+a*c^2+5*b*c^2-2*c^3,
    gap> #-5,     -2*a+c,     -2*a^2-a*b-2*b^2-3*a*c                                      
  
  
  A.2-4 Diff
  
  Diff( D, N )  operation
  Returns:  a homalg matrix
  
  If  D  is a f × p-matrix and N is a g × q-matrix then H=Diff(D,N) is an fg ×
  pq-matrix    whose   entry   H[g*(i-1)+j,q*(k-1)+l]   is   the   result   of
  differentiating N[j,l] by the differential operator corresponding to D[i,k].
  (Here we follow the Macaulay2 convention.)
  
    Example  
    gap> S := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c" * "x,y,z";;
    gap> D := HomalgMatrix( "[ \
    > x,2*y,   \
    > y,a-b^2, \
    > z,y-b    \
    > ]", 3, 2, S );
    <A 3 x 2 matrix over an external ring>
    gap> N := HomalgMatrix( "[ \
    > x^2-a*y^3,x^3-z^2*y,x*y-b,x*z-c, \
    > x,        x*y,      a-b,  x*a*b  \
    > ]", 2, 4, S );
    <A 2 x 4 matrix over an external ring>
    gap> H := Diff( D, N );
    <A 6 x 8 matrix over an external ring>
    gap> Display( H );
    2*x,     3*x^2, y,z,  -6*a*y^2,-2*z^2,2*x,0,  
    1,       y,     0,a*b,0,       2*x,   0,  0,  
    -3*a*y^2,-z^2,  x,0,  -y^3,    0,     0,  0,  
    0,       x,     0,0,  0,       0,     1,  b*x,
    0,       -2*y*z,0,x,  -3*a*y^2,-z^2,  x+1,0,  
    0,       0,     0,0,  0,       x,     1,  -a*x