(5 points) Let A € R"×k be a real matrix, and define S = AT A E R*xk. First, prove thatS is diagonalizable. Then, show that all the eigenvalues of

S are real and nonnegative.Finally, letting is dzS. k be the eigenvalues of S, prove that \forall x \in \mathbf{R}^{k}, \quad\|A x\| \leq \sqrt{\lambda_{k}}\|x\|