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

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

o3 = | 7/8 4/5 8/9 8/5 1/7 |
     | 1/2 4/7 8   9/5 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          -1/6774519 |, | 8281603080 
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/6774519 0          |  | -6412661710
                                                               | 243882684  
                                                               | -1479140495
                                                               | 234967999  
     ------------------------------------------------------------------------
     -8697240 32182920  14197680  -36386280 |)
     9724680  -24492055 -19103875 28043680  |
     0        0         0         0         |
     2205     -5435570  1723750   1726760   |
     -1212309 913507    714910    -939988   |

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

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

o6 = true