Exercise 6.7: Sub-Gaussian Random Matrices

chapter 6

(a)

Let $B=UD{U}^{\mathsf{\text{T}}}$ be the spectral decomposition of $B$. Then $$\begin{array}{}\text{(1)}& \mathbb{E}[e^{\lambda Q}]=U^{\mathsf{\text{T}}}\mathbb{E}[e^{\lambda gD}]U\le U^{\mathsf{\text{T}}}{e}^{\frac{1}{2}{\lambda }^2{\sigma }^2{D}^2}U=e^{\frac{1}{2}{\lambda }^2{\sigma }^2{U}^{\mathsf{\text{T}}}{D}^{2}U}=e^{\frac{1}{2}{\lambda }^2{\sigma }^2{B}^2}.\end{array}$$

(b)

Follows from (a), that $B\le b{I}_d$ almost surely, and that the matrix exponential is matrix monotone with respect to simultaneously diagonalisable matrices.