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