Let $A$ be an $n\times m$ matrix and $B$ be a $m\times m$ matrix, and the ranks of these two matrices are arbitrary. My question is: do we always have $\mathcal{R}(A'BA) = \mathcal{R}(A'B)$ ? (the symbol $'$ represents the transpose and $\mathcal{R}(.)$ represents the column space.)
It is obvious that it establishes when the matrix $B$ is symmetrical positive definite. How about the most general case (e.g., both two matrices to be rank deficient, the matrix $B$ is not symmetrical)?