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Kronecker :: rationalNormalForm

rationalNormalForm -- rational normal form of a matrix

Synopsis

Description

This function produces a matrix B in rational normal form, and invertible matrices P and Q such that P*Q = I and B = P*A*Q.
i1 : R = ZZ/101[x]

o1 = R

o1 : PolynomialRing
i2 : M = R^4

      4
o2 = R

o2 : R-module, free
i3 : A = random(M,M)

o3 = | -7  8   22  -20 |
     | -5  25  -41 23  |
     | -23 22  15  3   |
     | -47 -18 33  -30 |

             4       4
o3 : Matrix R  <--- R
i4 : factor det(x*id_M - A)

       4     3      2
o4 = (x  - 3x  + 29x  - 36x + 5)

o4 : Expression of class Product
i5 : (B,P,Q) = rationalNormalForm A

o5 = (| 3   1 0 0 |, | 0 32  20  4   |, | -40 39  -7  1 |)
      | -29 0 1 0 |  | 0 -39 -31 -43 |  | -12 -26 -5  0 |
      | 36  0 0 1 |  | 0 45  -8  12  |  | -37 -31 -23 0 |
      | -5  0 0 0 |  | 1 -15 34  2   |  | -47 -41 -47 0 |

o5 : Sequence
i6 : B - P*A*Q == 0

o6 = true
i7 : P*Q - id_M == 0

o7 = true

Ways to use rationalNormalForm :