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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 = | 17  -41 11  50  |
     | 19  19  -28 -37 |
     | -44 27  -34 28  |
     | -11 31  -23 23  |

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

       4      3      2
o4 = (x  - 25x  - 30x  - 15x - 44)

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

o5 = (| 25 1 0 0 |, | 0 -7  -16 6   |, | 3   -9 17  1 |)
      | 30 0 1 0 |  | 0 -2  -8  1   |  | -21 0  19  0 |
      | 15 0 0 1 |  | 0 -13 -23 -4  |  | -3  44 -44 0 |
      | 44 0 0 0 |  | 1 22  -37 -42 |  | 35  50 -11 0 |

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

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

o7 = true

Ways to use rationalNormalForm :