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

              2        5
o2 : Matrix QQ  <--- QQ

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

o3 = | 1/2 5/3 7/2 3/10 5/9 |
     | 3/4 8/7 5/6 2    4/9 |

              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 |, | -1/113334 0         |, | -62100944
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0         -1/226668 |  | -4098318 
                                                             | 8699544  
                                                             | 21627160 
                                                             | 1700010  
     ------------------------------------------------------------------------
     1424   -655200 -2254592 5831616  |)
     -13587 -32760  -148764  395325   |
     5856   56160   324768   -834624  |
     4790   241020  795160   -2178330 |
     0      0       0        0        |

o4 : Sequence

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

o5 = true

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

o6 = true