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

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

o3 = | 2   1 7/3 9/5 1/6 |
     | 3/4 9 5/6 7/3 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         -1/414811 |, | 715236   
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/414811 0         |  | -10533503
                                                             | -16143480
                                                             | 24831720 
                                                             | 12444330 
     ------------------------------------------------------------------------
     -166740 70656   -71592  19740   |)
     645     -98192  106256  -10525  |
     142470  -215310 433740  -298980 |
     225     255150  -541740 141030  |
     0       0       0       0       |

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

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

o6 = true