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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 = | 6   2  30  -49 |
     | -35 38 -6  16  |
     | -10 3  -32 7   |
     | 6   14 4   31  |

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

       4      3      2
o4 = (x  - 43x  + 26x  + 5x - 43)

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

o5 = (| 43  1 0 0 |, | 0 37 -4  24  |, | 23 -22 6   1 |)
      | -26 0 1 0 |  | 0 38 -28 -27 |  | 42 30  -35 0 |
      | -5  0 0 1 |  | 0 25 -49 -3  |  | 31 -5  -10 0 |
      | 43  0 0 0 |  | 1 37 -28 -17 |  | 12 -5  6   0 |

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

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

o7 = true

Ways to use rationalNormalForm :