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

              2        5
o2 : Matrix QQ  <--- QQ

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

o3 = | 3/4 3/5 5/7 2   9/5 |
     | 2/9 6/7 3/2 4/5 1/5 |

              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         -1/363005 |, | 1750212  
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/217803 0         |  | -6150886 
                                                             | 15246210 
                                                             | -29492748
                                                             | 28040677 
     ------------------------------------------------------------------------
     1216872 -3393180 523152  -220932 |)
     -1666   -472710  163464  19096   |
     0       0        0       0       |
     -290238 834120   -349398 68868   |
     -183988 644595   115752  -111832 |

o4 : Sequence

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

o5 = true

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

o6 = true