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

              2        5
o2 : Matrix QQ  <--- QQ

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

o3 = | 7  9/5 8/7 3/5 6/7 |
     | 10 1   1/2 1   7/4 |

              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 |, | -1/4145255 0          |, | 32529952  
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0          -1/4145255 |  | -91153944 
                                                               | 58033570  
                                                               | -9625728  
                                                               | -144878076
     ------------------------------------------------------------------------
     1231632  -319130  88398    -257136  |)
     -2126664 -2739765 -34866   2025912  |
     0        0        0        0        |
     1791432  1045520  -1271592 -757856  |
     -6846336 2791740  241416   -1623972 |

o4 : Sequence

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

o5 = true

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

o6 = true