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

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

o3 = | 1 4/9 2/3 3/5 3/4  |
     | 3 3   1/2 4/3 10/3 |

              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 |, | -1/23883 0         |, | -588100 -15580
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0        -1/238830 |  | -660510 27360 
                                                            | 477660  0     
                                                            | -752850 37905 
                                                            | 1353240 -25764
     ------------------------------------------------------------------------
     -3960  -142900 98520   |)
     -15120 -328500 50490   |
     0      0       0       |
     -86940 -97650  111195  |
     51948  463320  -250236 |

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

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

o6 = true