How to prove $A(A^TA)^{-1}A^T$ is a singular matrix

Question

Is there a way to prove that statement? Lets say A is a $n\times p$ matrix.

I read from my text book that this matrix is a projection matrix of rank p.

Have no idea what that means, and no luck searching for understandable literature.

Hope you guys can lend me a hand.

Thanks.

Answer 1

If $p<n$, $A(A^TA)^{-1}A^T \in \mathbb{R}^{n \times n}$, but $$\operatorname{rank}(A(A^TA)^{-1}A) \le \min(\operatorname{rank}(A(A^TA)^{-1}), \operatorname{rank}(A))\le p<n$$

Hence, it is singular.

However, if $p=n$, the statement is not necessarily true. For example, let $A=I$.