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 = | 2    5/6 1/10 2/5 2   |
     | 7/10 1/2 4/7  7/6 9/4 |

              2        5
o2 : Matrix QQ  <--- QQ

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

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

              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          -1/5001572 |, | 44836875  
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/1250393 0          |  | -234976020
                                                               | 350110040 
                                                               | -270670260
                                                               | 89698350  
     ------------------------------------------------------------------------
     1478740  1582665  -539875 -173300 |)
     579120   -8125236 559230  33600   |
     0        0        0       0       |
     -3427200 -1537200 1136580 -354300 |
     -1034600 2110290  -545650 230160  |

o4 : Sequence

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

o5 = true

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

o6 = true