How to find eigenvalues and eigenvectors of this matrix

Solution 1:

If no further constraints are imposed on the $\theta_k$, there seems to be no simple form for the eigenvalues( other than $0$, which has multiplicity $\geq 2$ as explained in the OP). Indeed, I show an example below where the nonzero eigenvalues are not expressible by radicals.

Let $M=7$, and choose the following values for the $\theta_k$ (note that they are between $\frac{\pi}{4}$ and $\frac{\pi}{2}$).

$$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline k & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \cos(\theta_k) & \frac{20}{29} & \frac{65}{97} & \frac{48}{73} & \frac{3}{5} & \frac{28}{53} & \frac{33}{65} & \frac{8}{17} \\ \hline \sin(\theta_k) & \frac{21}{29} & \frac{72}{97} & \frac{55}{73} & \frac{4}{5} & \frac{45}{53} & \frac{56}{65} & \frac{15}{17} \\ \hline \end{array} $$

Then we have

$$ C=\left(\begin{array}{ccccccc} \frac{-399}{1649} & \frac{132}{493} & 0 & 0 & 0 & 0 & \frac{75}{2813} \\ \frac{-119}{7081} & \frac{92}{2117} & \frac{-75}{2813} & 0 & 0 & 0 & 0 \\ 0 & \frac{-27}{365} & \frac{44}{485} & \frac{-119}{7081} & 0 & 0 & 0 \\ 0 & 0 & \frac{-23}{265} & \frac{620}{3869} & \frac{-27}{365} & 0 & 0 \\ 0 & 0 & 0 & \frac{-83}{3445} & \frac{36}{325} & \frac{-23}{265} & 0 \\ 0 & 0 & 0 & 0 & \frac{-47}{1105} & \frac{60}{901} & \frac{-83}{3445} \\ \frac{-47}{1105} & 0 & 0 & 0 & 0 & \frac{132}{493} & \frac{-427}{1885} \\ \end{array}\right) $$

The characteristic polynomial of $C$ is $\chi_C=X^2P$, where $$ P=x^5 - \frac{198395974}{60131320925}x^4 - \frac{58043714683396506772}{723155151237068571125}x^3 + \frac{12655356459504058576}{3615775756185342855625}x^2 + \frac{1065764359163841911}{723155151237068571125}x - \frac{186332188794386}{1701541532322514285} $$

And PARI-GP tells us that this polynomial is irreducible and has Galois group ${\mathfrak S}_5$, which is not solvable. So the roots of $P$ are not expressible by radicals.