I have a very complicated 4 by 4 matrices with lots of complex variables in each entry. I can divide the matrix into quarters such as
M = $\begin{pmatrix} M_1 & M_2 \\ M_2^\dagger & M_3 \end{pmatrix} $
where $M_1,M_2,M_3$ are all 2 by 2 hermitian matrices with known eigenvalues. Given this knowledge, How can I get the eigenvalues of matrix M?
(I tried it in Mathematica but it only returns 'Root[...]' with very complicated and lengthy arguments.)
Each matrix is given as
$M_1=$ \begin{equation} \left( \begin{array}{cc} 4 \phi ^2 |h_d|^2 |\lambda _{\mu }|^2+|\lambda_{\mu}|^2\phi ^4 +3|\lambda_{\nu}|^2|l|^4 -2m_{\text{LHu}}^2 & 6 A_{\nu }^* \lambda_{\nu}^*{l^*}^2 +6 \lambda _{\mu } \lambda_{\nu}^*\phi ^2 l^* h_d +18 |\lambda_{\nu}|^2 l^3 l^* \\ 6 A_{\nu}\lambda_{\nu}l^2 +6 \lambda _{\nu}\lambda _{\mu }^*\phi^2lh_d^*+18 |\lambda_{\nu}|^2|l|^2 {l^*}^2 & 4 \phi ^2|\lambda_{\mu}|^2 |h_d|^2 +|\lambda_{\mu}|^2\phi^4 +3|\lambda_{\nu}|^2|l|^4 -2m_{\text{LHu}}^2\\ \end{array} \right) \end{equation}
\begin{equation} M_2=\left( \begin{array}{cc} 4 l \phi ^2 h_d |\lambda_{\mu}|^2 & 4 |\lambda _{\mu}|^2\phi ^2 l h_d^* +3 \lambda _{\mu}\lambda_{\nu}^*\phi^2 \left(l^*\right)^2 \\ 4 |\lambda _{\mu}|^2\phi ^2 l^* h_d + 3\lambda_{\mu}\lambda_{\nu}^* \phi ^2 l^2 & 4|\lambda_{\mu}|^2 \phi ^2 l^*h_d^* \\ \end{array} \right) \end{equation}
\begin{equation} M_3=\left( \begin{array}{cc} -2 g^2 |h_d|^2 +|\lambda_{\mu}|^2\phi ^4 + m_{\text{Hd}}^2 & -g^2 {h_d^*}^2 \\ -g^2 h_d^2 & -2 g^2 |h_d|^2 +|\lambda_{\mu}|^2\phi ^4 + m_{\text{Hd}}^2 \\ \end{array} \right) \end{equation}