Solving a characteristic Polynomial of the Hilbert Matrix

Question

I need to find the eigenvalues of the following characteristic polynomial but I can't seem to successfully find the roots of the equation:

$P[λ]$ = $λ^5$ - $563/315λ^4$ + $0.3476λ^3$ - $0.0038λ^2$

Ive tried factoring this to find the multiple λ values but have had 0 luck solving for it.

Thanks

Answer 1

Factoring by $\lambda$, you get $P(\lambda) = \lambda^2 ( \lambda^3 - \frac{563}{315} \lambda^2 + 0.3476 \lambda - 0.0038)$. Therefore, $0$ is a double eigenvalue.

If you want an exact expression of the other eigenvalues, you can then use the root formula for a degree 3 polynomial (see this wikipedia article).

Otherwise, you may use a root-finding algorithm, such as Newton's method.

For future reference, you may use Wolfram|Alpha to make such computations for you (see this link for a Wolfram|Alpha solution). The other eigenvalues are 0.0116221, 0.208652 and 1.56703.