I have a matrix of the form \begin{equation} M = \begin{pmatrix} A_1 & a_{12}I & a_{13}I & \dots & a_{1n}I\\ a_{21}I & A_2 & a_{23}I & \dots & a_{2n}I\\ a_{31}I & a_{32}I & A_3 & \ddots & \vdots \\ \vdots & & \ddots & \ddots & a_{n-1,n}I\\ a_{n1} & & & a_{n,n-1}I & A_n \end{pmatrix}, \end{equation} where $I$ is an $m\times m$ identity matrix, $A_i$ are non-commuting $m\times m$-matrices and $a_{ij}$ are complex numbers.
I know the eigenvalues of all the $A_i$. Is there a way to find the eigenvalues of $M$ using this information?
I have tried to use the Schur complement to compute the determinant of $(M-\lambda I)$, but that does not really help since that includes inverting the $A_i$-matrices. Also, since the $A_i$ do not commute, it is not obvious to me if one can compute the determinant by considering the blocks as elements in an $n\times n$-matrix.