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 = | 6/7 3/7 10/3 9/5 3/10 |
     | 5/4 4   5/9  5/4 7/3  |

              2        5
o2 : Matrix QQ  <--- QQ

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

o3 = | 1   3/10 3/2 6/5 5/2 |
     | 8/5 9/2  5/9 2   8/7 |

              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         -1/321590 |, | 9183975 
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/192954 0         |  | -301580 
                                                             | 1157724 
                                                             | -6221135
                                                             | -1345890
     ------------------------------------------------------------------------
     1630500 -1496775 710325  -587775 |)
     -201552 105000   -45724  89556   |
     0       0        0       0       |
     -678012 641375   -421869 254391  |
     -302568 278250   -76146  25074   |

o4 : Sequence

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

o5 = true

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

o6 = true