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In the matrix product $A=BC$ let $A$, $B$, $C$ be squared real matrices with the same dimensions, where $A$ and $C$ are symmetrical and $C$ is also positive definite.

Is there any set of conditions that $B$ must verify (especially on its eigenvalues) so that $A$ is non-negative definite (resp. positive definite)?

Michael Galuza
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Danaroth
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  • Do you think there are? Have you done any work so far? – Robin Goodfellow Aug 12 '15 at 16:23
  • Well, the solution seems trivial if B is symmetrical as well, since I just need for it to be non-negative definite, so the eigenvalues must be non-negative. I am mostly interested in how to extend, if possible, the reasoning to the case B is in general non symmetrical. – Danaroth Aug 12 '15 at 16:43
  • Ok, I think I got the hang of it. I could only prove a necessary condition for $B$: its eigenvalues must be real non-negative (resp. positive). Since $C$ is definite positive, its inverse $C^{-1}$ exists and it's definite positive (similar question), so the problem is the same as finding conditions on eigenvalues of a product of a positive definite and a non negative definite (resp.positive definite) matrix (similar question) – Danaroth Aug 14 '15 at 06:33

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