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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   1/4 5/2 2/3 1/10 |
     | 1/2 2   2   8   1/5  |

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

o3 = | 4/5  3/2 1 2/5 4/9 |
     | 10/9 5   3 3/4 5/3 |

              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         -1/82027 |, | 802152   3762  
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/164054 0        |  | -2408966 -44   
                                                            | -507079  1386  
                                                            | 494316   -10824
                                                            | 7382430  0     
     ------------------------------------------------------------------------
     55152  13608  -379980 |)
     -16344 42228  -7330   |
     -22068 -77406 156325  |
     6156   7944   -13500  |
     0      0      0       |

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

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

o6 = true