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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 = | -31 11  -15 45  |
     | -24 31  48  -37 |
     | -2  4   -25 5   |
     | 25  -41 17  8   |

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

       4      3     2
o4 = (x  + 17x  - 8x  - 16x - 5)

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

o5 = (| -17 1 0 0 |, | 0 -38 -1  20  |, | -50 34  -31 1 |)
      | 8   0 1 0 |  | 0 -22 39  -18 |  | 18  -11 -24 0 |
      | 16  0 0 1 |  | 0 11  -6  -2  |  | -45 40  -2  0 |
      | 5   0 0 0 |  | 1 -3  -47 35  |  | 32  -29 25  0 |

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

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

o7 = true

Ways to use rationalNormalForm :