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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 = | 2/3 3   1 4/5 4/5 |
     | 1/4 1/3 6 1/8 5/4 |

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

o3 = | 5/8 6   9   1   1/6 |
     | 1/2 1/2 9/5 1/2 3/7 |

              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 |, | -1/6560 0       |, | -10905756 354300 
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | 0       -1/3280 |  | -693180   20700  
                                                         | 32800     0      
                                                         | 10486662  -338550
                                                         | 1159893   -42525 
     ------------------------------------------------------------------------
     -150624 -63216 -117012 |)
     -10080  -4080  -5460   |
     0       0      0       |
     145008  63192  103474  |
     18312   4788   14511   |

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

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

o6 = true