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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 = | -26 45  -39 34 |
     | -23 -42 48  4  |
     | -28 38  6   -8 |
     | -8  29  18  11 |

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

                       2
o4 = (x + 14)(x + 44)(x  - 7x + 29)

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

o5 = (| 1 0 0   0 |, | 13  47  -14 -19 |, | 9   -12 -18 -41 |)
      | 0 1 0   0 |  | 8   18  39  27  |  | -3  -11 -20 -12 |
      | 0 0 7   1 |  | -40 -18 -38 -47 |  | -43 -32 -9  1   |
      | 0 0 -29 0 |  | 43  12  -11 -3  |  | 39  36  -2  0   |

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

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

o7 = false

Ways to use rationalNormalForm :