Computing the spectral norm of a projection matrix

Question

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 $A$?

Answer 1

We recognize in that expression a projection matrix onto $\operatorname{Row(A)}$ and since $A$ is a full row rank we have that

$$P=A^T(AA^T)^{-1}A \implies \sup\left\{\frac{\|Ax\|}{\|x\|},\, x\neq 0\right\}=1$$

Answer 2

Given fat matrix $\rm A$ with full row rank,

$$\rm P := A^\top \left( A A^\top \right)^{-1} A$$

is the (symmetric) projection matrix that projects onto the row space of $\rm A$. Every projection matrix has eigenvalues $0$ and $1$. Since $\rm P$ is symmetric and positive semidefinite,

$$\| \rm P \|_2 = \sigma_{\max} (\rm P) = \lambda_{\max} (\rm P) = 1$$