6. Let $A$ be a Hermitian matrix with eigenvalues $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_n$ and orthonormal eigenvectors $\mathbf{u}_1, \ldots, \mathbf{u}_n$. For any nonzero vector $\mathbf{x}$ in $\mathbb{R}^n$, the Rayleigh quotient $\rho(\mathbf{x})$ is defined by

(a) If $\mathbf{x} = c_1\mathbf{u}_1 + \cdots + c_n\mathbf{u}_n$, show that

(10%)

(b) Show that $\lambda_n \le \rho(\mathbf{x}) \le \lambda_1$. (5%)

(c) Show that $\max_{\mathbf{x}\neq 0} \rho(\mathbf{x}) = \lambda_1$ and $\min_{\mathbf{x}\neq 0} \rho(\mathbf{x}) = \lambda_n$. (5%)