Take the 2-minute tour ×  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?  share|improve this question What do you mean by
positive definite for an asymmetric matrix? There are different
options. –  Simon Markett Sep 2 '12 at 13:40 I meant that M also
satisfies $x^T M x > 0$. –  User2012 Sep 2 '12 at 14:43 add
comment  1 Answer  It's not true in general: take $$
N=\begin{bmatrix}1&0\\0&1/5\end{bmatrix},\ \ \
M=\begin{bmatrix}1&1\\0&1\end{bmatrix}. $$ $M$ is positive-
definite according to your definition, since $$
\begin{bmatrix}x\\ y\end{bmatrix}^TM\begin{bmatrix}x\\
y\end{bmatrix}=x^2+xy