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getMultiplicationMatrix -- Computes the matrix over a $k$-basis for multiplication by an element in a finite dimensional $k$-algebra

Description

For an algebra C over a field $k$ or a polynomial ring S and an ideal I, this function generates a matrix with entries in $k$ representing multiplication by the user-prescribed element in C or S/I respectively.

We compute the multiplication matrix of the element 1+a*b+b*c+c*a in the algebra L[a,b,c]/(a^2,b^2,c^2), where L is the cyclotomic extension $\mathbb{Q}(\zeta_{7})$ represented as the quotient of the polynomial ring $\mathbb{Q}[x]$ by the ideal generated by the polynomial $x^6+x^5+x^4+x^3+x^2+x+1$. The multiplication matrix is computed over L with respect to the basis $\{1,a,b,c\}$.

i1 : L = QQ[x]/(x^6+x^5+x^4+x^3+x^2+x+1);
 
i2 : F = toField L;
 
i3 : getMultiplicationMatrix(F[a,b,c], ideal(a^2,b^2,c^2),1+a*b+b*c+c*a)

o3 = | 1 0 0 0 0 0 0 0 |
     | 0 1 0 0 0 0 0 0 |
     | 1 0 1 0 0 0 0 0 |
     | 0 1 0 1 0 1 0 1 |
     | 1 0 0 0 1 0 0 0 |
     | 0 0 0 0 0 1 0 0 |
     | 1 0 0 0 0 0 1 0 |
     | 0 0 0 0 0 0 0 1 |

             8      8
o3 : Matrix F  <-- F

Alternatively for a $k$-algebra C, we can compute the multiplication matrix of an element b with respect to a basis of C.

i4 : R = QQ[x,y]/(x^2+y^2+1, 3*x+2);
 
i5 : getMultiplicationMatrix(R, 1+y*x^2)

o5 = | 1   -52/81 |
     | 4/9 1      |

              2       2
o5 : Matrix QQ  <-- QQ

See also

Ways to use getMultiplicationMatrix:

  • getMultiplicationMatrix(Ring,Ideal,Thing)
  • getMultiplicationMatrix(Ring,Thing)

For the programmer

The object getMultiplicationMatrix is a method function.

The source of this document is in A1BrouwerDegrees/Documentation/TraceAndNormDoc.m2:43:0.