# On the positve definiteness of a particular matrix and inverse

I have the following question. Let $M=BAC=M^T$ where $B\in\mathbb{R}^{m\times n}$, $C\in\mathbb{R}^{n\times m}$ and $A\in\mathbb{R}^{n\times n}$ invertible. Suppose $M$ is positive definite, $B$ is full row rank, $C$ is full column rank and $B\neq C^T$.

Question: Is there any possible scenario (potential additional conditions for $A$, $B$ or $C$) where I can conclude that the matrix $W=BA^{-1}C$ is also positive definite as a consequence of the positive definiteness of $M$?

Thanks in advance!

Best,

N.

• Possible scenario? Well, yes. If $B=C^T$ with $BC$ of full rank, then $W$ is also going to be positive definite. Are you asking for "possible scenario" because you actually have a more concrete problem in which you want to reach that conclusion? Posting the original problem is always more effective. You never know what properties you are leaving out that might be useful. – user583012 Aug 10 '18 at 12:57
• @fatherBrown Thanks for the comment. I'll try to edit the question to be more precise. – Nico F. Aug 10 '18 at 13:00
• @fatherBrown I have just made a few edits, with more information about the matrices. The original problem is quite extensive, but it basically reduces to that case. Thanks again for your comment! – Nico F. Aug 10 '18 at 13:05