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

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

o3 = | 9/2 5/7 1/4 7/5 1/4 |
     | 2   5   1/5 10  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 |, | -1/766243 0         |, | -273252  
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0         -1/766243 |  | -10501869
                                                             | -84215280
                                                             | -3354390 
                                                             | 137923740
     ------------------------------------------------------------------------
     140856   -209376 732      25740  |)
     1016736  239708  -1212036 25865  |
     -2742960 513060  64080    -7740  |
     -481680  -88240  604590   -94550 |
     0        0       0        0      |

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

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

o6 = true