Eigenvalues of system with 5 or more degrees of freedom?

When finding eigenvalues for a system consisting of a single particle, its position and velocity are used when making the system of equations. So that there is an equation like
$dot{x} =begin{bmatrix}a & b c & dend{bmatrix}x$.

Where $x$ is the vector list of the position and velocity of the particle, i.e. $x=(q, dot{q})^text{T}$.

For particles with more degrees of freedom, how would one find the eigenvalues of the matrix when it is larger than $5$ x $5$. This would lead to atleast a sixth order polynomial. This question arises because I recall polynomials higher than 5th order are in general, not solvable exactly.

Are other analytical techniques used here?