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

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

o3 = | 6   1/3 4/5 1/9 3/8 |
     | 2/5 5   2/5 9/2 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          -1/5236165 |, | 38251920  
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/5236165 0          |  | -151084467
                                                               | -268764840
                                                               | 181380114 
                                                               | 41889320  
     ------------------------------------------------------------------------
     -389760   -23800   664440   -856800  |)
     -18158784 1418970  -1544304 3957435  |
     7778295   -32850   -4627080 -1182600 |
     19519668  -2735190 2068128  -4215870 |
     0         0        0        0        |

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

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

o6 = true