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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 = | 17  -46 22 49  |
     | -43 -12 42 -30 |
     | -50 42  36 -4  |
     | -32 30  12 -27 |

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

               3      2
o4 = (x + 10)(x  - 24x  - 7x - 7)

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

o5 = (| 1 0  0 0 |, | -34 25  27  6   |, | -7  -35 17  -26 |)
      | 0 24 1 0 |  | -11 17  -27 -16 |  | 36  38  -5  1   |
      | 0 7  0 1 |  | -4  -3  -44 -24 |  | -44 -30 29  0   |
      | 0 7  0 0 |  | -25 -43 22  22  |  | 10  14  -47 0   |

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

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

o7 = false

Ways to use rationalNormalForm :