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Kronecker :: kroneckerNormalForm(Matrix,Matrix)

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

Synopsis

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   2   8/9 1/6 4   |
     | 5/4 4/5 2/3 5/4 3/7 |

              2        5
o2 : Matrix QQ  <--- QQ
i3 : B = random(R^2, R^5)

o3 = | 3   3/2 1 1/6 1   |
     | 5/8 4   2 1/4 3/4 |

              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       -1/9876 |, | -89968  320  
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/6584 0       |  | -436490 1030 
                                                         | 899595  -1125
                                                         | -126264 -8280
                                                         | 46088   0    
     ------------------------------------------------------------------------
     3296   -3232  -1664 |)
     8140   -17810 4520  |
     -22698 37287  -8964 |
     3600   -5256  3552  |
     0      0      0     |

o4 : Sequence
i5 : P*A*Q - A' == 0

o5 = true
i6 : P*B*Q - B' == 0

o6 = true