Singular Book 2.1.7 -- maps induced by Hom

i1 : A = QQ[x,y,z]

o1 = A

o1 : PolynomialRing

i2 : M = matrix(A, {{1,2,3},{4,5,6},{7,8,9}})

o2 = | 1 2 3 |
     | 4 5 6 |
     | 7 8 9 |

             3       3
o2 : Matrix A  <--- A

i3 : Hom(M,A^2)

o3 = | 1 0 4 0 7 0 |
     | 0 1 0 4 0 7 |
     | 2 0 5 0 8 0 |
     | 0 2 0 5 0 8 |
     | 3 0 6 0 9 0 |
     | 0 3 0 6 0 9 |

             6       6
o3 : Matrix A  <--- A

i4 : Hom(A^2,M)

o4 = | 1 2 3 0 0 0 |
     | 4 5 6 0 0 0 |
     | 7 8 9 0 0 0 |
     | 0 0 0 1 2 3 |
     | 0 0 0 4 5 6 |
     | 0 0 0 7 8 9 |

             6       6
o4 : Matrix A  <--- A

Notice that the basis that Macaulay2 uses for Hom(A^3,A^2) is different than the basis used by Singular.

The function contraHom of the Singular book in example 2.1.7 could be coded in the following way.

i5 : contraHom = (M, s) -> (
         (n,m) := (numgens target M, numgens source M);
         R := mutableMatrix(ring M, s*n, s*m);
         for b from 0 to m-1 do
           for a from 0 to s-1 do
             for c from 0 to n-1 do
               R_(a*n+c,a*m+b) = M_(b,c);
         matrix R
         )

o5 = contraHom

o5 : FunctionClosure

Let's try an example.

i6 : contraHom(M,2)

o6 = | 1 4 7 0 0 0 |
     | 2 5 8 0 0 0 |
     | 3 6 9 0 0 0 |
     | 0 0 0 1 4 7 |
     | 0 0 0 2 5 8 |
     | 0 0 0 3 6 9 |

             6       6
o6 : Matrix A  <--- A