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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 = | -30 -17 12 7   |
     | 3   -27 41 -5  |
     | 27  20  8  11  |
     | 12  -23 -7 -25 |

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

                       2
o4 = (x - 47)(x - 10)(x  + 30x - 12)

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

o5 = (| 10 0  0   0 |, | -47 -39 6   50  |, | -21 26  -8 7   |)
      | 0  47 0   0 |  | -21 2   3   -41 |  | -18 -14 37 -29 |
      | 0  0  -30 1 |  | -20 1   -33 38  |  | 47  37  21 1   |
      | 0  0  12  0 |  | -28 21  -2  -15 |  | 1   1   37 0   |

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

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

o7 = true

Ways to use rationalNormalForm :