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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  -32 39 16  |
     | 37  9   39 2   |
     | -10 -44 38 -26 |
     | -43 35  -9 3   |

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

                       2
o4 = (x - 24)(x - 18)(x  - 25x - 10)

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

o5 = (| 1 0 0  0 |, | -31 2  -3  38  |, | -36 -21 17  1  |)
      | 0 1 0  0 |  | 45  47 -36 -18 |  | -37 44  27  17 |
      | 0 0 25 1 |  | -11 12 9   50  |  | 21  -41 -13 1  |
      | 0 0 10 0 |  | 9   -1 9   29  |  | -21 -28 38  0  |

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

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

o7 = false

Ways to use rationalNormalForm :