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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 = | -22 12 33  -45 |
     | 49  1  -4  27  |
     | -28 15 16  -44 |
     | -1  22 -22 33  |

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

       4      3      2
o4 = (x  - 28x  - 17x  + 17x - 50)

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

o5 = (| 28  1 0 0 |, | 0 37 -31 -46 |, | 37  -9  -22 1 |)
      | 17  0 1 0 |  | 0 49 24  12  |  | -31 -35 49  0 |
      | -17 0 0 1 |  | 0 44 47  31  |  | 13  38  -28 0 |
      | 50  0 0 0 |  | 1 40 -27 -33 |  | 8   -34 -1  0 |

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

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

o7 = true

Ways to use rationalNormalForm :