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

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

o3 = | 1/2 1/2 3/4 4/3 3/2 |
     | 9/5 9/4 3/2 4   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         -1/396633 |, | -8539860
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/132211 0         |  | 2337984 
                                                             | -1606804
                                                             | 2932056 
                                                             | 264422  
     ------------------------------------------------------------------------
     -2007180 -104160 170380 620760  |)
     527040   -136980 -76608 10296   |
     -275832  350200  -8456  -270244 |
     710208   -106560 -30408 -183792 |
     0        0       0      0       |

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

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

o6 = true