Suppose that \(A\) is an \(m\)-by-\(n\) matrix with complex entries. Prove that \(A^*A\) is a Hermitian matrix with all nonnegative eigenvalues. Hint: Show that \(\langle x, A^*Ax \rangle \ge 0\) for all \(x \in \mathbb{C}^n\). What if \(x\) is an eigenvector of length 1?
If \(A\) is a normal matrix with eigenvalues \(\lambda_1, \ldots, \lambda_n\) and corresponding orthonormal basis of eigenvectors \(x_1, \ldots, x_n\), prove that \(A = \lambda_1 x_1 x_1^* + \ldots + \lambda_n x_n x_n^*\).