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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 3/5 4 1 1 |
     | 1 2   4 2 2 |

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

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

              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 |, | -1/8370 0       |, | -5840  -3020
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0       -1/2790 |  | 5350   2050 
                                                         | 8370   0    
                                                         | -43095 1635 
                                                         | 23925  -2175
     ------------------------------------------------------------------------
     -96    614   -360  |)
     -600   350   -2250 |
     0      0     0     |
     -15912 -762  3105  |
     16560  -1290 -675  |

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

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

o6 = true