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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 = | 35 -9  44  -50 |
     | 46 -41 -43 7   |
     | 33 -3  43  -8  |
     | 21 3   7   7   |

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

                      2
o4 = (x - 5)(x + 23)(x  + 39x + 18)

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

o5 = (| 1 0 0   0 |, | 15  -30 -11 -45 |, | -49 9   49  -42 |)
      | 0 1 0   0 |  | 35  48  11  44  |  | -1  -16 -25 -18 |
      | 0 0 -39 1 |  | -48 11  0   10  |  | -18 0   24  1   |
      | 0 0 -18 0 |  | -8  36  10  8   |  | -22 -20 -20 0   |

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

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

o7 = false

Ways to use rationalNormalForm :