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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 = | -13 18  23  10  |
     | -26 -11 -7  -22 |
     | 39  7   35  35  |
     | -35 -46 -44 -7  |

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

       4     3      2
o4 = (x  - 4x  + 27x  + 9x + 33)

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

o5 = (| 4   1 0 0 |, | 0 36  -18 34 |, | -34 46  -13 1 |)
      | -27 0 1 0 |  | 0 -8  17  22 |  | -9  10  -26 0 |
      | -9  0 0 1 |  | 0 -35 5   20 |  | -42 -44 39  0 |
      | -33 0 0 0 |  | 1 26  -16 0  |  | 20  -22 -35 0 |

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

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

o7 = true

Ways to use rationalNormalForm :