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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 10/7 7/4 8    7/5 |
     | 9/7 1    4/5 10/9 9   |

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

o3 = | 1/3 5/4 1/10 5/6 1/2 |
     | 7   8/7 1/9  3/2 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 |, | 0           -1/23850383 |, |
      | 0 0 0 1 0 |  | 0 0 0 0 1 |  | -1/71551149 0           |  |
                                                                 |
                                                                 |
                                                                 |
     ------------------------------------------------------------------------
     -230227242  -848253   -3450888 1244712   10078572  |)
     -887974388  -2583280  1245468  6865768   -62303780 |
     -9541343160 -37288440 5600160  5295060   68080320  |
     2139706764  8688834   -1159866 -11431944 -4606776  |
     715511490   0         0        0         0         |

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

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

o6 = true