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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 1 9/10 9/5 4   |
     | 3/7 2 2/7  5   9/8 |

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

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

              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 |, | -1/7976287 0          |, | -524081320
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0          -1/7976287 |  | 366538170 
                                                               | 558340090 
                                                               | -122959720
                                                               | -47286400 
     ------------------------------------------------------------------------
     3097080  -13112400 -733376  15852872  |)
     -2665560 15114570  1548456  -28560399 |
     0        0         0        0         |
     1392960  -5248500  -2337440 10279920  |
     -2632000 1451520   825160   -953680   |

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

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

o6 = true