Exercise 1.8.2

The Rayleigh quotient $R(x)=\frac{x^T\mathit{Sx}}{x^{T}x}=\frac{{\lambda }_1{c}_1^2+\cdots +{\lambda }_n{c}_n^2}{c_1^2+\cdots +c_n^2}=\frac{{\lambda }_1\left(c_1^2+\cdots +c_n^2\right)+\left(({\lambda }_2-{\lambda }_1)c_2^2+\cdots +({\lambda }_n-{\lambda }_1)c_n^2\right)}{c_1^2+\cdots +c_n^2}={\lambda }_1+\frac{({\lambda }_2-{\lambda }_1)c_2^2+\cdots +({\lambda }_n-{\lambda }_1)c_n^2}{c_1^2+\cdots +c_n^2}$

If ${\lambda }_1\ge {\lambda }_i$ for $i=2,3,\dots ,n$, then we see that the second term is less than or equal to zero. The maximum value is ${\lambda }_1$ and it’s achieved when $c_2=\cdots =c_n=0$ and $c_1=1$. (Actually, $c_1$ doesn’t have to be $1$ here, any nonzero constant is fine)