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

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

o3 = | 6/5 1   5/7 1/2 5/4 |
     | 1/2 2/9 6/5 6   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 |, | -1/1797448 0          |, | -736552860
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0          -1/3594896 |  | 731495712 
                                                               | 251642720 
                                                               | -12264735 
                                                               | -16995770 
     ------------------------------------------------------------------------
     -3443100 831960   -758240  662580  |)
     4154400  -3785280 3072384  -438624 |
     0        0        0        0       |
     150465   -115290  366408   -630819 |
     -78330   837700   -1876560 -32850  |

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

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

o6 = true