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 = | 9/7 3 1/2 3 2/3  |
     | 5/2 3 4   1 9/10 |

              2        5
o2 : Matrix QQ  <--- QQ

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

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

              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 |, | -1/339443 0          |, | -18114474
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0         -1/1018329 |  | 1178995  
                                                              | 14256606 
                                                              | 10596930 
                                                              | -28749060
     ------------------------------------------------------------------------
     -21042  146244  -571830 149940  |)
     80787   -140731 -30250  34110   |
     0       0       0       0       |
     -183330 85095   201000  25830   |
     -7140   -31680  334440  -558900 |

o4 : Sequence

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

o5 = true

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

o6 = true