Consider the matrix $$ A=\left[\begin{array}{cc} H_{1} & 0\\ 0 & H_{2} \end{array}\right]-\left[\begin{array}{cc} R_{1} & R_{2}\\ R_{1} & R_{2} \end{array}\right] $$ where $H_1$ and $H_2$ are negative definite and symmetric matrices over the reals, and where $R_1$ and $R_2$ are symmetric matrices over the reals. Notice the block nature of $A$.
I would like to diagonalize $A$, that is finding an invertible matrix $M$ such that $M^{-1}KM$ is diagonal. I am hoping that there is a way to take advantage of $K$'s specific structure to do this. Any help would be much appreciated.