Eigenvalues in Python: A Bug?

As far as I know, assumption 1 is correct, but assumption 2 is not.

A Real Symmetric matrix produces eigenvalues and eigenvectors that are real only.

However, for a given eigenvalue, the associated eigenvector isn't necessarily unique.

Furthermore, round-off error shouldn't be so significant for a matrix that actually isn't that big, or contain numbers that aren't very small.

For comparison, I ran your test matrix through a JavaScript version of RG.F (Real General, from the EISPACK Library): Eigenvalues and Eigenvectors Calculator

Here is the output:

Eigenvalues:

   20   12   20   20   20   20   20   20

Eigenvectors:

 0.9354143466934854     0.35355339059327395     -0.021596710639534     -0.021596710639534     -0.021596710639534     -0.021596710639534     -0.021596710639533997     -0.021596710639533997-0.1336306209562122     0.3535533905932738     -0.15117697447673797     -0.15117697447673797     -0.15117697447673797     -0.15117697447673797     -0.15117697447673797     -0.15117697447673797-0.1336306209562122     0.3535533905932738     0.9286585574999623     -0.15117697447673797     -0.15117697447673797     -0.15117697447673797     -0.15117697447673797     -0.15117697447673797-0.1336306209562122     0.3535533905932738     -0.15117697447673797     0.9286585574999623     -0.15117697447673797     -0.15117697447673797     -0.15117697447673797     -0.15117697447673797-0.1336306209562122     0.3535533905932738     -0.15117697447673797     -0.15117697447673797     0.9286585574999623     -0.15117697447673797     -0.15117697447673797     -0.15117697447673797-0.1336306209562122     0.3535533905932738     -0.15117697447673797     -0.15117697447673797     -0.15117697447673797     0.9286585574999623     -0.15117697447673797     -0.15117697447673797-0.1336306209562122     0.3535533905932738     -0.15117697447673797     -0.15117697447673797     -0.15117697447673797     -0.15117697447673797     0.9286585574999622     -0.15117697447673797-0.1336306209562122     0.3535533905932738     -0.15117697447673797     -0.15117697447673797     -0.15117697447673797     -0.15117697447673797     -0.15117697447673797     0.9286585574999622

No imaginary components.

To confirm, or deny, the validity of results, you could always write a small program that plugs the results back into the original equation. Simple matrix and vector multiplication. Then you'd know for sure whether or not the outputs are correct. Or, if they are wrong, how far away from correct answers they are.