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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 = | 21  -5  -24 2   |
     | -9  -49 -23 8   |
     | -43 -5  7   15  |
     | 24  -15 21  -36 |

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

               3     2
o4 = (x - 46)(x  + 2x  + 6x + 34)

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

o5 = (| 1 0   0 0 |, | 12  -37 12  -43 |, | 24  -42 -16 -39 |)
      | 0 -2  1 0 |  | 50  31  8   -50 |  | -11 -17 -1  1   |
      | 0 -6  0 1 |  | 34  13  38  47  |  | -26 -50 -45 0   |
      | 0 -34 0 0 |  | -13 -1  -16 -38 |  | 11  -4  -42 0   |

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

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

o7 = false

Ways to use rationalNormalForm :