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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 = | -3  20  24 12 |
     | -50 -18 -3 2  |
     | 38  18  11 12 |
     | -40 37  -8 28 |

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

                       2
o4 = (x + 27)(x + 47)(x  + 9x - 31)

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

o5 = (| 1 0 0  0 |, | -22 -21 38  34  |, | 7   12  47  13 |)
      | 0 1 0  0 |  | -47 -26 25  50  |  | 20  -44 -22 -7 |
      | 0 0 -9 1 |  | -40 -17 -3  -28 |  | -33 -48 -25 1  |
      | 0 0 31 0 |  | 0   -20 -38 30  |  | -15 -43 21  0  |

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

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

o7 = false

Ways to use rationalNormalForm :