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 1/2  10/3 2/5 3/4 |
     | 2 7/10 3/2  5/8 3   |

              2        5
o2 : Matrix QQ  <--- QQ

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

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

              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 |, | -1/1022351 0          |, | -20541510
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0          -1/1022351 |  | -31937620
                                                               | 6134106  
                                                               | 10271800 
                                                               | 15939440 
     ------------------------------------------------------------------------
     -1835415 39663   374760  206622  |)
     -453800  22590   269120  -425160 |
     0        0       0       0       |
     175760   -111480 -79000  143040  |
     1292880  -8488   -636960 -68344  |

o4 : Sequence

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

o5 = true

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

o6 = true