Definiteness of A'BA

Question

Let $A$ be a $(k \times n)$ matrix and $B$ a $(k \times k)$ matrix. In that case, is there a general result for the definiteness of the $(n \times n)$ matrix $A'BA$? If not, what if $B$ is known to be positive definite. Can the definiteness of $A'BA$ then be determined?

Best,

Esben

Answer 1

I assume your matrices are real. $A'BA$ has rank at most $k$, so if $k < n$ it certainly can't be positive definite or negative definite. If $B$ is positive semidefinite, $A'BA$ is positive semidefinite, since for any vector $x$ we have $$ x' A' B A x = (A x)' B (A x) \ge 0$$