Let $\mathbf \Psi$ denote the set of all positive definite, diagonal, nXn dimensional, real-valued matrices . Let $\mathbf \Phi$ denote the set of all positive semi-definite, diagonal, nXn dimensional, real-valued matrices . Let $\mathbf \Pi$ denote the set of all linear transformations defined by a nXn dimensional matrix of Real numbers; Let $ \mathbf \Pi$ exclude the Zero Matrix.
Proposition 1: $\; \forall \; \Psi \in \mathbf \Psi $, and $\; \forall \; \Pi \in \mathbf \Pi \;$, the matrix product $\; \Pi \Psi \;$ is positive definite.
Proposition 2: $\; \forall \; \Phi \in \mathbf \Phi $, and $\; \forall \; \Pi \in \mathbf \Pi \;$, the matrix product $\; \Pi \Phi \;$ is positive semi-definite.
If Proposition 1 is false then please indicate what subset of $\Pi$ are such that $\; \Pi \Psi \;$ is positive definite.
If Proposition 2 is false then please indicate what subset of $\Pi$ are such that $\; \Pi \Phi \;$ is positive semi-definite.