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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 = | 48  45  42 -17 |
     | -8  18  34 -15 |
     | -22 18  24 5   |
     | 37  -37 19 43  |

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

       4      3
o4 = (x  - 32x  + 20x - 38)

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

o5 = (| 32  1 0 0 |, | 0 27 24 1   |, | 20  -13 48  1 |)
      | 0   0 1 0 |  | 0 17 38 29  |  | 36  -13 -8  0 |
      | -20 0 0 1 |  | 0 18 21 -30 |  | -50 -28 -22 0 |
      | 38  0 0 0 |  | 1 29 16 -21 |  | 27  13  37  0 |

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

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

o7 = true

Ways to use rationalNormalForm :