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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 = | -34 0   21  0  |
     | -33 -29 -39 13 |
     | 4   45  -28 34 |
     | 27  -11 -30 28 |

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

                       2
o4 = (x + 25)(x + 30)(x  + 8x + 28)

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

o5 = (| 1 0 0   0 |, | 34  13  -13 -12 |, | 40 33  7   42 |)
      | 0 1 0   0 |  | -13 -3  33  -3  |  | -8 -27 -1  31 |
      | 0 0 -8  1 |  | -13 -11 -22 -43 |  | 46 -37 20  1  |
      | 0 0 -28 0 |  | -43 17  -33 -15 |  | 4  37  -45 0  |

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

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

o7 = false

Ways to use rationalNormalForm :