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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   1/6 10/9 1/3 8   |
     | 1/3 5/8 6/5  3   5/4 |

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

o3 = | 1/9 9   4/5 1 8/5 |
     | 7   7/4 1/5 7 4   |

              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 |, | -1/732598 0         |, | -18394416
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0         -1/366299 |  | 873448   
                                                             | -40870350
                                                             | 15157504 
                                                             | 7325980  
     ------------------------------------------------------------------------
     -223317 5229   108576  -41589 |)
     24192   -83472 16936   -1872  |
     -531720 3150   -66990  41145  |
     232461  15549  -110896 -11447 |
     0       0      0       0      |

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

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

o6 = true