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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 7 4/5 8/7 1 |
     | 2 2 1   7   5 |

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

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

              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 |, | -1/401001 0         |, | 1599324 
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0         -1/401001 |  | -236766 
                                                             | -829530 
                                                             | -4280808
                                                             | 5614014 
     ------------------------------------------------------------------------
     -12530 -585495 82900  31224  |)
     -70105 60993   2336   4440   |
     118230 -19980  -9345  -79560 |
     6720   152712  -80304 1176   |
     0      0       0      0      |

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

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

o6 = true