Chapter 7.1, Problem 26E

Let $A$ be an $\left(n\times n\right)$ symmetric matrix and consider the function $R$ defined on $R^n$ by

$R\left(x\right)=\frac{x^{T}Ax}{x^{T}x}$, $x\ne \theta$.

The number $R\left(x\right)$ is called a Rayleigh quotient. Let $A$ have eigenvalues ${\lambda }_1$, ${\lambda }_2$, …, ${\lambda }_n$, where ${\lambda }_1\le {\lambda }_2\le {\lambda }_3\le \mathrm{...}\le {\lambda }_n$. Prove that for $x$ in $R^n$, ${\lambda }_1\le R\left(x\right)\le {\lambda }_n$. [Hint: By the corollary to Theorem $23$ in Section $4.7$, $R^n$ has an orthonormal basis $\left\{u_1,u_2,\mathrm{...},u_n\right\}$, where $A{u}_i={\lambda }_i{u}_i$, $1\le i\le n$. For a given $x$, $x\ne \theta$, we can express $x$ as $x=a_1{u}_1+a_2{u}_2+\mathrm{...}+a_n{u}_n$. Using this expansion, calculate $x^{T}Ax$ and $x^{T}x$.

COROLLARY Let $A$ be an $\left(n\times n\right)$ symmetric matrix. It is possible to choose eigenvectors $u_1,u_2,\mathrm{...},u_n$ for $A$ such that $\left\{u_1,u_2,\mathrm{...},u_n\right\}$ is an orthonormal basis for $R^n$.