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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 = | 4/5 10/3 9/5 1   7   |
     | 4/3 3    4/9 4/3 1/4 |

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

o3 = | 1/8 1 1/5 2/9 5/3 |
     | 1/4 1 1/2 9/8 7/8 |

              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 |, | -1/270974 0         |, | -15717000
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0         -1/135487 |  | 8080695  
                                                             | 12193830 
                                                             | -5709492 
                                                             | -4371636 
     ------------------------------------------------------------------------
     -389880 1541520 -78240 113760  |)
     187635  -805680 -22896 23316   |
     0       0       0      0       |
     -17100  238752  25092  -166152 |
     -81060  173376  16260  -368    |

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

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

o6 = true