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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 = | 44  3   49  -38 |
     | -25 50  28  42  |
     | 9   -22 22  36  |
     | 0   23  -22 -13 |

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

       4     3     2
o4 = (x  - 2x  + 8x  + 12x + 14)

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

o5 = (| 2   1 0 0 |, | 0 -32 -44 25  |, | 38  -21 44  1 |)
      | -8  0 1 0 |  | 0 7   -3  30  |  | 29  23  -25 0 |
      | -12 0 0 1 |  | 0 42  27  -4  |  | -22 33  9   0 |
      | -14 0 0 0 |  | 1 20  17  -43 |  | -46 35  0   0 |

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

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

o7 = true

Ways to use rationalNormalForm :