I want to convert the inverse of sum of two hermitian matrices into sum of two or more matrices. I mean I want to simplify the bellow equation in a way that not to have inverse of sum of matrices any more. In the bellow equation $B$ is a $L\times L$ diagonal matrix with real elements on the diagonal and $A$ is a $N\times L$ matrix.
$(A^HA+B^{-2})^{-1}$
Thanks in advance.
We can write $$ (B^{-2} + A^HA)^{-1} = (B^{-1})^{-1}(I + B A^HA B)^{-1}(B^{-1})^{-1} = B(I + B A^HA B)^{-1}B $$ So now, it suffices to calculate $(I + B A^HA B)^{-1}$. If you'd prefer an infinite sum, we can write (if the entries of $B$ are sufficiently small) $$ (I + B A^HA B)^{-1} = I - B A^HA B + (B A^HA B)^2 - (B A^HA B)^3 + \cdots $$ Note also that $I + B A^HA B$ is Hermitian.