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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 = | 1/3 6/7 2/3 8/5 3/10 |
     | 1   1/2 1/2 5/4 5/8  |

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

o3 = | 3/2 1/2 1   1/8 8   |
     | 5/9 1/2 4/7 1/3 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         -1/142033 |, | -1136976
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/142033 0         |  | -6078184
                                                             | 5965386 
                                                             | 1039120 
                                                             | -168840 
     ------------------------------------------------------------------------
     -153180 -98208  139125 12888  |)
     208544  -343434 -92050 8624   |
     0       0       0      0      |
     -82992  197544  -64820 -3432  |
     16984   36792   -19320 -20656 |

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

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

o6 = true