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 = | 10/3 3 3/8 9/4 6   |
     | 7/9  1 7   1/2 5/6 |

              2        5
o2 : Matrix QQ  <--- QQ

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

o3 = | 4   1   6/7  4/5 1/3 |
     | 1/4 8/9 1/10 5/3 1/8 |

              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 |, | -1/634806 0         |, | 52952697  
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0         -1/211602 |  | -453324126
                                                             | 29624280  
                                                             | 223689340 
                                                             | 111508878 
     ------------------------------------------------------------------------
     -16578  -184275 61587   18468   |)
     187644  111060  -471018 39096   |
     0       0       0       0       |
     -85720  -36120  233440  -152640 |
     -158268 60390   113754  27432   |

o4 : Sequence

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

o5 = true

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

o6 = true