Computing the spectral norm of a projection matrix I was reading a paper in which there was an argument as trivial, but could not make myself sure about it. It is said that given a full row-rank matrix $A$, the norm (probably $\ell_2$-induced matrix norm) of $A^T(AA^T)^{-1}A$ is one. Is that trivial, and correct for any given matrix $A$?

We recognize in that expression a projection matrix onto $\operatorname{Row(A)}$ and since $A$ is a full row rank we have that
$$P=A^T(AA^T)^{-1}A \implies \sup\left\{\frac{\|Ax\|}{\|x\|},\, x\neq 0\right\}=1$$
