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 = | 1/9 9/8 6/7  7/3 7/8 |
     | 1/2 5/8 9/10 5/6 10  |

              2        5
o2 : Matrix QQ  <--- QQ

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

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

              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           -1/16285627 |, | -9651447  
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/32571254 0           |  | -964135284
                                                                 | 1139993890
                                                                 | 64241730  
                                                                 | -47211900 
     ------------------------------------------------------------------------
     -1285704 16230375   -6796080  -13944420 |)
     16213344 -169816080 90298320  59477964  |
     0        0          0         0         |
     -7008330 79924635   -56963340 -27747405 |
     -1993584 3141600    -556896   -707868   |

o4 : Sequence

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

o5 = true

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

o6 = true