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

Synopsis

Usage:
(A',B',P,Q) = kroneckerNormalForm(A,B)
Function: kroneckerNormalForm
Inputs:
- A, a matrix
- B, a matrix
Optional inputs:
- ChangeMatrix => ..., -- 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 = | 1/7 3   5    1/2 5   |
     | 3/4 5/7 3/10 3/2 5/2 |

              2        5
o2 : Matrix QQ  <--- QQ

i3 : B = random(R^2, R^5)

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

              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           -1/134272658 |, |
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/67136329 0            |  |
                                                                  |
                                                                  |
                                                                  |
     ------------------------------------------------------------------------
     -265575114 8802360    -53505900 3246390   990486    |)
     64291829   59251640   -11030355 -77763    -16877511 |
     -156222815 -24070760  5047350   -13586475 9534735   |
     -90370800  -117317200 30996030  1131130   5634720   |
     134272658  0          0         0         0         |

o4 : Sequence

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

o5 = true

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

o6 = true