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, a matrix
- B, a matrix
Optional inputs:
- ChangeMatrix => ..., default value {true, true}, Kronecker normal form of a matrix of linear forms
Outputs:
- A', a matrix
- B', a matrix
- P, a matrix
- Q, a matrix

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   51928   25914  -37872 -1740  |)
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/132721 0         |  | 202908   121152  -76944 -3708  -7740  |
                                                             | 2079708  135492  -10962 -8932  -31530 |
                                                             | -7142692 -427840 44982  36652  37850  |
                                                             | 4777956  0       0      0      0      |

o4 : Sequence

i5 : P*A*Q - A' == 0

o5 = true

i6 : P*B*Q - B' == 0

o6 = true