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

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

o3 = | 4/3 6/7 3/2  7/5 3/5 |
     | 9/5 9/2 1/10 1   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         -1/8481033 |, | -130780680
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/942337 0          |  | -18141270 
                                                              | 44018240  
                                                              | 15802960  
                                                              | 169620660 
     ------------------------------------------------------------------------
     -760380  315000   -1042422 494760   |)
     746760   -2136330 386652   -189210  |
     2325140  455520   637856   -1018430 |
     -2224250 519900   72640    62720    |
     0        0        0        0        |

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

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

o6 = true