Suppose $A$ is a positive definite matrix and $B$ is a non-symmetric
matrix with all positive principal minors. 


Is their product $AB$ a matrix with all positive principal minors?


I believe the answer is yes, and I have been trying to find a proof but got stuck along the way. The wiki page for minor gives a corollary to the Cauchy-Binet formula which I think may be of use:
<http://en.wikipedia.org/wiki/Minor_%28linear_algebra%29#Applications>


Thank you,
Alex



Principal Minors of Matrix Product
This isn't true even if $A$ and $B$ are both symmetric and positive definite. For example, let $$A=\begin{pmatrix} 1 & 2\\2 & 5\end{pmatrix}, \quad B=\begin{pmatrix} 1 & -2\\-2 & 5\end{pmatrix},\quad\text{then}\quad AB=\begin{pmatrix} -3 & 8\\-8 & 21\end{pmatrix}.$$


