Does a projection matrix have to be a square matrix?

Question

Does a projection matrix have to be a square matrix? I know that it's computed by a formula $$P = A(A^TA)^{-1}A^T$$ where $A$ can be virtually of any dimensions. Does this formula guarantee that $P$ will be a square matrix?

Answer 1

Say $A$ is $m\times n$. Then $A^T$ is $n\times m$. In particular, $A^TA$ is the product of $n\times m$ by $m\times n$, giving an $n\times n$ matrix. The inverse is also an $n\times n$ matrix. So $P$ is a product of $$(m\times n)(n\times n)(n\times m)$$ matrices, and so $P$ will necessarily be an $m\times m$ matrix. Thus, the formula yields a square matrix whether or not $A$ is a square matrix.