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

              2        5
o2 : Matrix QQ  <--- QQ

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

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

              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          -1/2970062 |, | 1129202270 
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/2970062 0          |  | -284386080 
                                                               | -24112515  
                                                               | -1271244275
                                                               | 623713020  
     ------------------------------------------------------------------------
     6061860  84980   -1704990 -715400  |)
     -2219016 -542478 601290   603480   |
     -941178  -111867 -78525   427980   |
     169750   1040725 312375   -2596090 |
     0        0       0        0        |

o4 : Sequence

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

o5 = true

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

o6 = true