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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 = | 3/10 3/10 2   1/10 1   |
     | 5    1    1/5 3/2  7/3 |

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

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

              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 |, | -1/13138921 0           |, | 6756858   
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0           -1/13138921 |  | -543857568
                                                                 | -19782385 
                                                                 | 36111680  
                                                                 | 197083815 
     ------------------------------------------------------------------------
     2368530   -6752970 4063770   -726264  |)
     -13461600 36283800 -36396720 8285544  |
     -4992750  -4374825 4783870   -985320  |
     1745000   -1095990 1321450   -2971440 |
     0         0        0         0        |

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

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

o6 = true