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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 = | 1/2 3/2 2   5/7 5/8 |
     | 7/9 3   3/4 8/5 7/5 |

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

o3 = | 3    8/5 1 1/3 5/2 |
     | 7/10 7/4 1 9/2 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 |, | -1/34095119 0           |, |
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0           -1/34095119 |  |
                                                                 |
                                                                 |
                                                                 |
     ------------------------------------------------------------------------
     -1031310450 5516910   -13254300 5360400   -1799820 |)
     -184375200  2290120   2791110   -17242200 5560300  |
     112416200   -21202552 946680    10313100  -371520  |
     -398989290  2962890   765975    3579660   -9376500 |
     1363804760  0         0         0         0        |

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

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

o6 = true