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

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

o3 = | 1/10 9/7 7/6 1/4 1/2 |
     | 4/5  1/3 1   5/3 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 |, | 0          -1/2730369 |, | -515228560
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/2730369 0          |  | -238497000
                                                               | 256785171 
                                                               | 37730058  
                                                               | 98293284  
     ------------------------------------------------------------------------
     -1504720 -8417430 3055850  646800   |)
     -2252040 -3102120 2404080  -1592640 |
     2707656  4007745  -2861940 -737100  |
     -451920  617922   -230460  450324   |
     0        0        0        0        |

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

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

o6 = true