Consider the matrix $E$ where $E$ is defined as follows:
$E=A+\gamma BC+DB-BC^{-1}DC$
where $\gamma$>0, $A,C$ are positive definite, $D$ is Hurwitz and $B$ is positive semidefinite. I have a conjecture that $E$ is nonsingular. Am I correct? One way I was trying to approach the problem is to explore the eigenvalues of $E$. Any counter-examples which can disprove the conjecture might be enough if the conjecture is not true.
I was trying to start as follows. We know that $\lambda(XZ)=\lambda(ZX)$ where $\lambda(.)$ denotes the eigenvalue of $(.)$. Using this result, we can conclude that $\lambda(C^{-1}DC)=\lambda(CC^{-1}D)=\lambda(D)$ and somehow proceed from here. However I am not able to figure out a way to move from here.