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

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

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

              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         -1/506644 |, | -4601793 
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/253322 0         |  | -10941840
                                                             | 759966   
                                                             | 1704900  
                                                             | 12848550 
     ------------------------------------------------------------------------
     290052  41370  -265629 -162120 |)
     460740  15645  -633780 73710   |
     0       0      0       0       |
     -69048  -63770 107912  5580    |
     -789420 -14105 749910  -15450  |

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

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

o6 = true