Let $K_p$ be a positive definite hermitian matrix diagonnaly dominant (fyi: it is a stiffness matrix and each element is real), $K_n$ another positive definite hermitian matrix diagonnaly dominant (fyi: it is a stiffness matrix and each element is real), and $L$ a diagonal matrix. Each element of $L$ is a positive real number included in $[0,1]$.
Is the product $LK^{-1}_nK_p$ positive definite ? positive semi-definite ? Neither of them ? If so, what are the conditions on $L$ to have $LK^{-1}_nK_p$ positive definite or positive semi-definite.
According to me, $LK^{-1}_nK_p$ is not positive definite or positive semi-definite as the matrices cannot commute (am I right ?). For the rest, I really reach stalemate.
Thank you in advance for your help. EM