kroneckerNormalForm(Matrix,Matrix) -- normal form of a pair of matrices of scalars

Function: kroneckerNormalForm
Usage:

(A',B',P,Q) = kroneckerNormalForm(A,B)
Inputs:
- A, a matrix
- B, a matrix
Optional inputs:
- ChangeMatrix => ..., default value {true, true}, 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 = | 9/2 3/4 7/9  7/10 7   |
     | 9/4 7/4 7/10 7/3  3/7 |

              2       5
o2 : Matrix QQ  <-- QQ

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

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

              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 |, | 0           -1/25262993 |, |
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/10826997 0           |  |
                                                                 |
                                                                 |
                                                                 |
     ------------------------------------------------------------------------
     2654657740   -3952676  -8516004  3084480   1169756   |)
     17773707042  -42696528 -17330418 17580240  7728588   |
     -37768385025 56896560  64238265  -46815300 -10943100 |
     -4754499255  7937940   1938195   -2114910  -3641490  |
     1061045706   0         0         0         0         |

o4 : Sequence

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

o5 = true

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

o6 = true

Ways to use this method:

kroneckerNormalForm(Matrix,Matrix) -- normal form of a pair of matrices of scalars

The source of this document is in Kronecker.m2:1239:0.