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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 = | -13 -25 26  42  |
     | -34 -19 -10 5   |
     | 45  17  -30 -50 |
     | -5  32  12  33  |

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

       4      3      2
o4 = (x  + 29x  - 19x  - 15x + 24)

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

o5 = (| -29 1 0 0 |, | 0 26  -49 -32 |, | -48 -41 -13 1 |)
      | 19  0 1 0 |  | 0 19  11  -10 |  | 47  7   -34 0 |
      | 15  0 0 1 |  | 0 22  -12 25  |  | -12 -41 45  0 |
      | -24 0 0 0 |  | 1 -10 -37 -5  |  | -35 -42 -5  0 |

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

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

o7 = true

Ways to use rationalNormalForm :