Estimation of the eigenvalues of a matrix

Question

Given the following matrix \begin{pmatrix}0 & 1 & 0 & 0\\
-k & -\lambda_1 & -k & 0\\
0 & 0 & 0 & 1\\
-k & 0 & -k & -\lambda_2\end{pmatrix} where $k,\lambda_1$ and $\lambda_2$ are strictly positive, is there a way to estimate the eigenvalues, or at least say if they are positive or negative, complex or real? One eigenvalue is zero, but to find the others you need to solve a parametric cubic equation, which is not easy. Note that this matrix is the representation of a mechanical system, so a physical context may come in handy.

Answer 1

The eigenvalues of the matrix are the roots of the characteristic polynomial.

1. Compute the characteristic polynomial. Mathematica tells me that it is: $$p(s) = s^4+s^3(\lambda_1+\lambda_2)+s^2(\lambda_1\lambda_2+2k)+s k(\lambda_1+\lambda_2)=s q(s)$$ with $$q(s) = s^3 +s^2(\lambda_1+\lambda_2)+s(\lambda_1\lambda_2+2k)+ k(\lambda_1+\lambda_2)$$
2. Apply Routh's Stability Criterion to $q(s)$.

Routh's array should be as follows: $$\begin{matrix} s^3 & 1 & \lambda_1\lambda_2+2k\\ s^2 & \lambda_1+\lambda_2 & k(\lambda_1+\lambda_2)\\ s & \lambda_1\lambda_2+k\\ s^0 & k(\lambda_1+\lambda_2) \end{matrix}.$$ All the eigenvalues have negative real part if and only if the first column has only positive numbers (which is the case, given your assumptions).