<s>[INST] Is it possible to estimate the sign of real part of eigenvalues of a 10 by 10 matrix only by observing all the entries? I have a symbolic 10 by 10 matrix. It is not difficult to get the eigenvalue expressions by using Matlab. But the expressions of some eigenvalues are too long to be analyzed. I was wondering if there is a way to analyze the sign of real part of eigenvalues only by observing the matrix since all the entry expressions are pretty simple? 
The matrix is like this:
\begin{bmatrix}  -k_1-p_1\lambda & 0 &0 &0 & 0 &0 &0 &-\lambda^2p_1 &0 &0\\
                 0 &  -k_2-p_2\lambda & a_{13}\omega &0 & 0 &0 &0 &0 &-\lambda^2p_2 &0 \\
0  &a_{12}\omega & -k_3-p_3\lambda &0 & 0 &0 &0 &0 &0  &-\lambda^2p_3\\
-\frac{\sin{\theta}}{2} &0 &0 &0 &-\frac{\omega}{2} &0 &0 &0 &0 &0\\
\frac{\cos{\theta}}{2} &0 &0 &\frac{\omega}{2} &0 &0 &0 &0 &0 &0\\
0 &\frac{\cos{\theta}}{2} &-\frac{\sin{\theta}}{2} &0 &0 &0 &\frac{\omega}{2}&0 &0 &0\\
0 &\frac{\sin{\theta}}{2} &\frac{\cos{\theta}}{2} &0 &0 &-\frac{\omega}{2}&0 &0 &0 &0\\
-1 &0 &0 &0 &0 &0 &0 &-\lambda &0 &0\\
0 &-1 &0 &0 &0 &0 &0 &0 &-\lambda &0 \\
0 &0 &-1 &0 &0 &0 &0 &0 &0  &-\lambda
\end{bmatrix}.
Thanks.
 [/INST] Arrange the rows and columns in the order $1,8,4,5,2,3,9,10,6,7$. Your matrix is permutation-similar to the direct sum of
$$
A=\left[\begin{array}{cc|cc}
-k_1-p_1\lambda  &-\lambda^2p_1 &0 &0\\
-1 &-\lambda &0 &0\\
\hline
-\frac{\sin{\theta}}{2} &0 &0 &-\frac{\omega}{2}\\
\frac{\cos{\theta}}{2} &0 &\frac{\omega}{2} &0\\
\end{array}\right]
$$
and
$$
B=\left[\begin{array}{cccc|cc}
-k_2-p_2\lambda & a_{13}\omega &-\lambda^2p_2 &0 &0 &0\\
a_{12}\omega & -k_3-p_3\lambda &0  &-\lambda^2p_3 &0 &0\\
-1 &0 &-\lambda &0 &0 &0\\
0 &-1 &0 &-\lambda &0 &0\\
\hline
\frac{\cos{\theta}}{2} &-\frac{\sin{\theta}}{2} &0 &0 &0 &\frac{\omega}{2}\\
\frac{\sin{\theta}}{2} &\frac{\cos{\theta}}{2} &0 &0 &-\frac{\omega}{2}&0
\end{array}\right]
$$
The eigenvalues of the three $2\times2$ subblocks should be easy to analyse. It remains to find the eigenvalues of the $4\times4$ subblock of $B$.
</s>