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

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

o3 = | 4   3/2 1/2 2 1/2 |
     | 9/8 4/3 8   1 2/3 |

              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           -1/60737245 |, |
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/60737245 0           |  |
                                                                 |
                                                                 |
                                                                 |
     ------------------------------------------------------------------------
     2699430624  50093568  1035864  9854400   -30990528 |)
     -8599468560 -86614920 -1277760 -71596800 63557520  |
     854082954   12175128  -7623936 6501600   -4364298  |
     381690096   -38269728 792576   32363400  -14964632 |
     1822117350  0         0        0         0         |

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

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

o6 = true