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

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

o3 = | 9/5 4/3 3/8 3 7/9 |
     | 4   4   5/6 4 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        -1/139812 |, | -2855790 52920 
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/34953 0         |  | -1395252 -71856
                                                            | 3435792  225024
                                                            | 1333213  -27944
                                                            | 2202039  0     
     ------------------------------------------------------------------------
     -73170 3510   9990   |)
     -14067 -486   22815  |
     25560  -13608 -60240 |
     46959  -189   -20255 |
     0      0      0      |

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

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

o6 = true