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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 -10 7   -49 |
     | -14 25  -36 -49 |
     | 20  47  -28 -23 |
     | 42  -27 -49 -37 |

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

       4      3      2
o4 = (x  - 17x  - 31x  - 33x - 31)

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

o5 = (| 17 1 0 0 |, | 0 36  -14 -15 |, | 6   -44 -44 1 |)
      | 31 0 1 0 |  | 0 35  -32 -26 |  | -3  13  -14 0 |
      | 33 0 0 1 |  | 0 -33 31  44  |  | -44 -34 20  0 |
      | 31 0 0 0 |  | 1 -27 40  -27 |  | -47 36  42  0 |

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

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

o7 = true

Ways to use rationalNormalForm :