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.
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 $B$.