Chapter 6.6, Problem 11E

Let A be a symmetric $n\times n$ matrix with eigenvalues ${\lambda }_1,...,{\lambda }_n.$ Show that there exists an orthonormal set of vectors $\left\{x_1,...,x_n\right\}$ such that
$x^{T}Ax={\displaystyle \underset{i=1}{\overset{n}{\sum }}{\lambda }_i{\left(x^T{x}_i\right)}^2}$
for each $x\in {ℝ}^n.$