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

              2        5
o2 : Matrix QQ  <--- QQ

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

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

              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        -1/57616 |, | -299600  
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/28808 0        |  | -10944565
                                                           | 53735    
                                                           | 810640   
                                                           | 2592720  
     ------------------------------------------------------------------------
     18480   -8400 -17920 21840  |)
     -182895 38940 3850   3185   |
     -14555  396   19570  -20475 |
     30768   1728  -19360 -16016 |
     0       0     0      0      |

o4 : Sequence

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

o5 = true

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

o6 = true