Eigenvector does not work in Sympy

Question

I want to find eigenvectors of a matrix in Sympy and wrote the following program but it doesn't work. On the other hand, the A.eigenvects() function in Sympy calculates the eigenvalues and eigenvects of the matrix A and I used a similar thing here but when I want to print the results, an empty list is shown. Would you please guide me?

from sympy import *

H=Matrix([  [215.0   ,-104.1   ,5.1    ,-4.3    ,4.7    ,-15.1   ,-7.8],
        [-104.1  , 220.0   ,32.6   , 7.1    ,5.4    , 8.3    ,0.8],
        [ 5.1    , 32.6    ,  0.   , -46.8  , 1.0   , -8.1   , 5.1 ],
        [ -4.3   , 7.1     ,-46.8  ,125.0   ,-70.7  ,-14.7   ,-61.5],
        [ 4.7    , 5.4     , 1.0   ,-70.7   ,450.0  ,89.7    ,-2.5],
        [-15.1   , 8.3     ,-8.1   ,-14.7   ,89.7   ,330.0   ,32.7],
        [-7.8    ,0.8      ,5.1    ,-61.5   ,-2.5   ,32.7    ,280.0]])


zz=H.eigenvects()
pprint(zz)

Answer 1

The problem stems from the use of Floats and Rationals. eigenvects recasts Floats to Rationals. The eigenvals routine fails when roots fails to factor the polynomial. When you call eigenvals directly, the Floats are also recast but you can select not to recast them; when you do, roots is able to return values and the eigen values are returned.

>>> H.eigenvals()
{}
>>> H.eigenvals(rational=False)
{513.317044781366: 1, 101.965215714556: 1, 332.004505895816: 1, 268.369453977695: 1, 120.955771704237: 1, -23.7383543150805: 1, 307.126362241411: 1}

It seems like roots should be returning RootOf instances for this 7th order polynomial rather than returning no roots. It would be good to report this as an issue for SymPy:

roots(p) -> {} instead of RootOf instances when p = 5000000*x**7 - 8100000000*x**6 + 5146847850000*x**5 - 1623625381660000*x**4 + 261251048199624000*x**3 - 18813344309673222800*x**2 + 248990094307079384205*x + 20562650438939697400552