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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 = | 27  37 46 15  |
     | -1  -8 -6 -45 |
     | -23 22 23 20  |
     | 44  42 -6 20  |

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

                      2
o4 = (x - 42)(x - 4)(x  - 16x + 7)

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

o5 = (| 1 0 0  0 |, | -32 30  -31 8  |, | 7   -49 -8  -14 |)
      | 0 1 0  0 |  | -9  -19 24  26 |  | 38  -2  -2  -24 |
      | 0 0 16 1 |  | -39 8   50  24 |  | -9  -36 19  1   |
      | 0 0 -7 0 |  | -26 -32 -20 50 |  | -12 -25 -14 0   |

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

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

o7 = false

Ways to use rationalNormalForm :