kroneckerNormalForm(Matrix,Matrix) -- normal form of a pair of matrices of scalars

Synopsis

• Function: kroneckerNormalForm
• Usage:
(A',B',P,Q) = kroneckerNormalForm(A,B)
• Inputs:
• A,
• B,
• Optional inputs:
• ChangeMatrix => ..., default value {true, true}, Kronecker normal form of a matrix of linear forms
• Outputs:
• A',
• B',
• P,
• Q,

Description

This function gives the normal form of a pair of matrices (A,B) over a field of the same dimensions up to multiplication on either side by an invertible matrix. The return values are such that P*A*Q=A' and P*B*Q=B'.
 i1 : R = QQ o1 = QQ o1 : Ring i2 : A = random(R^2, R^5) o2 = | 9/2 9/4 1 3/2 7/4 | | 1/2 1/2 3/4 3/4 7/9 | 2 5 o2 : Matrix QQ <--- QQ i3 : B = random(R^2, R^5) o3 = | 7/10 7/10 7 5/2 2/3 | | 1/2 7/3 3/7 6/7 1 | 2 5 o3 : Matrix QQ <--- QQ i4 : (A',B',P,Q) = kroneckerNormalForm(A,B) o4 = (| 0 1 0 0 0 |, | 0 0 1 0 0 |, | 0 -1/132721 |, | -40808 | 0 0 0 1 0 | | 0 0 0 0 1 | | -1/132721 0 | | 202908 | 2079708 | -7142692 | 4777956 ------------------------------------------------------------------------ 51928 25914 -37872 -1740 |) 121152 -76944 -3708 -7740 | 135492 -10962 -8932 -31530 | -427840 44982 36652 37850 | 0 0 0 0 | o4 : Sequence i5 : P*A*Q - A' == 0 o5 = true i6 : P*B*Q - B' == 0 o6 = true