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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 = | 6/7 1/3 3/8 4/5 1 |
     | 5/7 3/8 2/3 10  5 |

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

o3 = | 7/10 1   3   2/3 1/2 |
     | 9/5  5/8 5/6 4/7 7/3 |

              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 |, | -1/3977077 0          |, | -73738210
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0          -1/7954154 |  | 202162464
                                                               | -51760176
                                                               | -10794630
                                                               | 23862462 
     ------------------------------------------------------------------------
     21348600  996450   -2138500 -24677380 |)
     -99078240 -1488528 10838592 91216944  |
     27972360  -1074528 -2924064 -24644256 |
     325710    56280    -854175  -15015    |
     0         0        0        0         |

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

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

o6 = true