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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:

Thank you, Alex

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If you know first $B$ is total positive, then it is easy for you to find counterexample yourself. – Sunni Sep 17 '10 at 3:18
up vote 5 down vote accepted

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

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Thank you. This is probably why I couldn't finish my proof :) – Alex Lupsasca Sep 12 '10 at 3:43
It's always good to start with $2\times 2$ case :) – Victor Protsak Sep 12 '10 at 4:56

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