next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
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 = | 10/3 5/8 1/2 9   7/6 |
     | 1    2   5/2 9/2 5   |

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

o3 = | 10/3 5/4 10/7 2/3  1/2 |
     | 2/5  3/2 4    10/9 2/3 |

              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 |, | -1/848271 0          |, | 4319100  
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0         -1/1696542 |  | -20704708
                                                              | 6817517  
                                                              | -1200357 
                                                              | 5089626  
     ------------------------------------------------------------------------
     -73530  -234360 375660   -49005  |)
     361040  -176876 -1656256 1071240 |
     -104062 63217   599480   -817950 |
     -86310  95571   -57420   -10800  |
     0       0       0        0       |

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

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

o6 = true