Linear functions on the classical matrix groups

Let $M$ be a random matrix in the orthogonal group ${\text{O}}_n$, distributed according to Haar measure, and let $A$ be a fixed $n\times n$ matrix over $\mathbb{R}$ such that $\text{tr}(A{A}^t)=n$. Then the total variation distance of the random variable $\text{tr}(AM)$ to standard normal is bounded by $2\sqrt{3}\mathrm{/}(n-1)$, and this rate is sharp up to the constant. Analogous results are obtained for $M$ a random unitary matrix and $A$ a fixed $n\times n$ matrix over $\text{C}$. The proofs are applications of a new abstract normal approximation theorem which extends Stein’s method of exchangeable pairs to situations in which continuous symmetries are present.