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

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

o3 = | 5/6 2 9/5 7/2 1   |
     | 2   2 7/9 5/4 1/2 |

              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 |, | 0          -1/1879429 |, | -15289164
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/1879429 0          |  | 24658864 
                                                               | -61971030
                                                               | 16050516 
                                                               | 18794290 
     ------------------------------------------------------------------------
     -1317744 -723924 349146  592290  |)
     2268504  -667488 -478856 -351120 |
     -4990680 1565100 157815  847620  |
     1584096  -251124 109340  -913280 |
     0        0       0       0       |

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

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

o6 = true