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

              2        5
o2 : Matrix QQ  <--- QQ

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

o3 = | 4/3 4/3 2   3/2 4/3 |
     | 9   10  5/9 1/5 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 |, | -1/1413236 0         |, | 21072682 
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0          -1/353309 |  | -16032645
                                                              | 25438248 
                                                              | -4820680 
                                                              | -37774144
     ------------------------------------------------------------------------
     -413960 3115752  -295182 -82460 |)
     398655  -2804220 258075  38700  |
     0       0        0       0      |
     261000  -1448280 -32040  44480  |
     -278320 257856   73152   -6280  |

o4 : Sequence

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

o5 = true

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

o6 = true