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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 = | -47 -27 30 15  |
     | 20  34  11 -35 |
     | 41  -32 -3 -24 |
     | -19 21  50 -42 |

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

       4      3      2
o4 = (x  - 43x  - 15x  + 2x - 34)

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

o5 = (| 43 1 0 0 |, | 0 11 -8  -11 |, | 16  -12 -47 1 |)
      | 15 0 1 0 |  | 0 27 40  35  |  | -46 48  20  0 |
      | -2 0 0 1 |  | 0 6  27  22  |  | -48 -12 41  0 |
      | 34 0 0 0 |  | 1 26 -42 14  |  | -2  20  -19 0 |

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

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

o7 = true

Ways to use rationalNormalForm :