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

              2        5
o2 : Matrix QQ  <--- QQ

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

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

              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          -1/1148498 |, | -2962302 
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/4593992 0          |  | -35493759
                                                               | 96473832 
                                                               | -1920870 
                                                               | -3800706 
     ------------------------------------------------------------------------
     87402   -494578 374360   267498   |)
     188145  34731   -1114740 -208971  |
     0       0       0        0        |
     -498270 -28250  73400    1753290  |
     -8034   75816   120120   -1862406 |

o4 : Sequence

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

o5 = true

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

o6 = true