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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 = | 7/10 7/2 1 4/5  7/5 |
     | 4    1   1 1/10 1   |

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

o3 = | 1/3 8   2/3 1    10  |
     | 3/4 7/9 2/3 10/3 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 |, | -1/588280 0         |, | -714840
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0         -1/294140 |  | -924882
                                                             | 3236259
                                                             | -402970
                                                             | 588280 
     ------------------------------------------------------------------------
     257600   -87360  -89600 -15120 |)
     63000    -101934 -2730  7812   |
     -1108300 456153  66235  62586  |
     149000   -47790  7550   -99180 |
     0        0       0      0      |

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

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

o6 = true