I have the following question:
Matrix $N$ is a diagonal matrix with all entries strictly positive (hence, $N$ is positive definite since it satisfies $x^T N x > 0$). Matrix $M$ is an asymmetric positive definite matrix with all entries non-negative.
Since $NM \neq MN$, it does not follow that the product $NM$ is positive definite. However, given the special structure of $N$, can we still show that $NM$ is positive definite? Or maybe, under certain additional conditions?