Exercise 6.13: Bernstein's Inequality for Vectors

chapter 6

(a)

Follows from $\mathrm{rank}(V_n)=\mathrm{rank}(\mathrm{\Sigma })+1$.

(b)

Simply note that $$\begin{array}{}\text{(1)}& \mathrm{tr}(V_n)=2\mathrm{tr}(\mathrm{\Sigma }),\Vert V_n{\Vert }_2=\mathrm{tr}(\mathrm{\Sigma }).\end{array}$$