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

              2        5
o2 : Matrix QQ  <--- QQ

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

o3 = | 1   10/3 3/7 4/7 3/5 |
     | 3/4 9/2  9   8/9 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 |, | -1/279426924 0           |, |
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0            -1/15523718 |  |
                                                                  |
                                                                  |
                                                                  |
     ------------------------------------------------------------------------
     -957026736 -286920000 -73964016 3867840  3168792  |)
     -175443480 86652000   -71903160 -466920  -746640  |
     -464604400 -20610880  39097800  335500   -1617920 |
     112687686  12098160   30551661  -4296645 23454    |
     2794269240 0          0         0        0        |

o4 : Sequence

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

o5 = true

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

o6 = true