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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 = | -10 33  18  17  |
     | 23  -45 24  16  |
     | -31 -33 -13 -35 |
     | 13  17  20  47  |

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

       2              2
o4 = (x  + 49x + 42)(x  - 28x - 4)

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

o5 = (| -49 1 0  0 |, | 28 3   29  -48 |, | -38 -31 -26 49 |)
      | -42 0 0  0 |  | 42 -29 -47 -49 |  | -43 44  47  38 |
      | 0   0 28 1 |  | 12 -42 -2  -25 |  | 1   1   42  1  |
      | 0   0 4  0 |  | 32 -28 2   36  |  | -39 0   -10 0  |

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

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

o7 = true

Ways to use rationalNormalForm :