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

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

o3 = | 1/8 1   7/5 2/5  9/4 |
     | 3/4 8/3 1/8 10/3 6/5 |

              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          -1/1257714 |, | 277845600 
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/1257714 0          |  | -215569848
                                                               | 150925680 
                                                               | 116617515 
                                                               | -34268360 
     ------------------------------------------------------------------------
     3408720  -230400 -1196280 -909840 |)
     -2417100 423720  195552   651360  |
     0        0       0        0       |
     906150   -642375 128295   -31110  |
     723800   -61320  -43260   -792400 |

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

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

o6 = true