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

              2        5
o2 : Matrix QQ  <--- QQ

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

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

              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 |, | -1/254535 0         |, | 737155  
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0         -1/509070 |  | -706344 
                                                             | -251160 
                                                             | -1293699
                                                             | 1527210 
     ------------------------------------------------------------------------
     -16495 -124700 -86160 564025  |)
     49596  -155580 42408  -219120 |
     -37110 11100   5220   -22650  |
     6501   141270  -24372 -412095 |
     0      0       0      0       |

o4 : Sequence

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

o5 = true

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

o6 = true