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

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

o3 = | 1 5    1   2/3 1 |
     | 2 7/10 3/4 1/6 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 |, | -1/9248279 0          |, | 4647160  
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0          -1/9248279 |  | 9454095  
                                                               | -56731090
                                                               | -6652236 
                                                               | 9248279  
     ------------------------------------------------------------------------
     -119700  -101290  4965730   -6294400 |)
     195300   -2268495 1633050   534780   |
     633780   2483310  -15583700 4122100  |
     -2235870 -431736  3679080   -752400  |
     0        0        0         0        |

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

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

o6 = true