Open source software for calculation of eigenvalues of symbolic matrix I have following matrix
\begin{bmatrix}
-\alpha & 0 & \beta & \gamma\cdot\omega_m \\
0 & -\alpha & -\gamma\cdot\omega_m & \beta \\
R_r\frac{L_h}{L_r} & 0 & -\frac{R_r}{L_r} & -\omega_m \\
0 & R_r\frac{L_h}{L_r} & \omega_m & -\frac{R_r}{L_r} 
\end{bmatrix}
where
$$ \alpha = \frac{R_s + R_r\frac{L^2_h}{L^2_r}}{L_{s\sigma}+\frac{L_h}{L_r}L_{r\sigma}} $$
$$ \beta = \frac{R_r\frac{L_h}{L^2_r}}{L_{s\sigma}+\frac{L_h}{L_r}L_{r\sigma}} $$
$$ \gamma = \frac{\frac{L_h}{L_r}}{L_{s\sigma}+\frac{L_h}{L_r}L_{r\sigma}}\cdot p_p $$
and I would like to calculate the eigenvalues of that in symbolic manner.
EDIT:
The matrix can be rewritten in following form
\begin{bmatrix}
-a & 0 & b & c\cdot d \\
0 & -a & -c\cdot d & b \\
e\cdot f & 0 & -e & -d \\
0 & e\cdot f & d & -e 
\end{bmatrix}
I have been looking for some open source software usable for that purpose. I have already tried the wxMaxima but I have received some overcomplicated expressions containing the square roots which I am not able to simplify. Can anybody recommend me any open source software which offers good results for eigenvalues calculation in symbolic manner?

This response is (perhaps) barely appropriate as an answer rather than a comment.  However, it may well be the best that the OP can do.
First of all, consider trying to programmatically identify the general roots of a quartic equation.  Although the general formula is somewhat unwieldy, writing a computer program (e.g. using java, c, python, ...) to calculate the roots should be very straight forward.
Similarly, writing a computer program to calculate the eigenvalues of a 4x4 matrix should also be straight forward.  Given the other responses to this posting, I would say (as a retired professional programmer) that the OP's surrendering to the need to write his own software may be best.
Edit 
It just occurred to me that dealing with roots like $(1 + \sqrt{2})$ or $[1 + \sin(23^{\circ})]$ may be problematic if the OP needs exactness rather than (for example) the right answers correct to 10 decimal places.
If exactness is needed, then the OP has to (somehow) anticipate all the various forms that the solution may come in and develop special methods to handle them.  For example, computing 
$(1 + \sqrt{2}) \times (3 - \sqrt{2}) \;=\; [1 + 2\sqrt{2}]$ would probably require special code.
