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

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

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

              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        -1/96818 |, | -2423976 44928 
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/48409 0        |  | 2964654  -22122
                                                           | -1139334 -33408
                                                           | -3484422 36846 
                                                           | 871362   0     
     ------------------------------------------------------------------------
     47400  -14640 -131400 |)
     30190  -16065 20480   |
     -53865 29505  60470   |
     -28620 -35280 -30960  |
     0      0      0       |

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

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

o6 = true