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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 2/5  4/3 3   5/6 |
     | 1   1/10 3/5 3/5 3/4 |

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

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

              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         -1/190903 |, | 295596  
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/190903 0         |  | 1648080 
                                                             | -8914920
                                                             | 2420490 
                                                             | 4581672 
     ------------------------------------------------------------------------
     -174260 271824  -113520 -54740  |)
     590800  -535080 421680  -305200 |
     -137655 -701820 395745  166110  |
     11450   337960  -276825 -24010  |
     0       0       0       0       |

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

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

o6 = true