Say I have 5 matrices call them $$A,B,C,D \text{ and } E.$$ Let $A,B$ and $C$ be $r\times k$ matrices which are fixed. Let $D$ and $E$ be $k\times k$ matrices which are free but fixed to be hermitian. I wish to know whether it is possible to provide general conditions on when there exists a solution to the matrix equation of the form A+BD+CE=0, i.e. under what conditions can $D$ and $E$ can be chosen in this manner.
I see by example that there sometimes exist solutions to this equation and sometimes don't. Which I suspect depends on the relative size of $r$ and $k$, i.e. The number of free parameters versus the number of constraints. However before diving into this I wanted to check if general conditions on solutions of this form are known?