Exercise 6.7: Sub-Gaussian Random Matrices

chapter 6

(a)

Let $B = U D U^\T$ be the spectral decomposition of $B$. Then \begin{equation} \E[e^{\lambda Q}] = U^\T \E[e^{\lambda g D}] U \le U^\T e^{\tfrac12 \lambda^2 \sigma^2 D^2} U = e^{\tfrac12 \lambda^2 \sigma^2 U^\T D^2 U} = e^{\tfrac12 \lambda^2 \sigma^2 B^2}. \end{equation}

(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.