If $C$ is nonsingular, show that $A$ and $C^TAC$ have the same rank.

Question

The proof goes like: $rank(C^TAC) \le rank(A)$ and $rank(C^TAC) \ge rank\bigl((C^T)^{-1}C^TACC^{-1}\bigr)$. So $rank(C^TAC) = rank(A)$.

Why is $rank(C^TAC) \le rank(A)$??

Answer 1

Because $\operatorname{rank}AB\leqslant\operatorname{rank}A$ and $\operatorname{rank}AB\leqslant\operatorname{rank}B$. This follows from the fact that$$\operatorname{rank}A=\dim\operatorname{Im}A.$$

So,$$\operatorname{rank}C^TAC\leqslant\operatorname{rank}AC\leqslant\operatorname{rank}A.$$