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

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

o3 = | 5/3 7/3 1/6 5 2/3 |
     | 5/2 2/7 2   2 8   |

              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         -1/7233588 |, | -149266284
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/602799 0          |  | 183184680 
                                                              | -28566846 
                                                              | -42494355 
                                                              | 57868704  
     ------------------------------------------------------------------------
     -4099620 -3536708 633552  129488  |)
     -7661640 2624104  281400  34496   |
     1321110  491274   -506520 13482   |
     4897935  -62055   -325620 -180270 |
     0        0        0       0       |

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

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

o6 = true