Let $A$ be a fixed $n$ by $n$ positive definite matrix with eigenvalues $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_n > 0$, and let $f(A):=\sum_{i=1}^n\log(\lambda_i)$, and let $X$ be a random $n$ by $k$ matrix with iid $N(0,\sigma^2 I)$ entries.

Question

• How close is $f(A+XX^T)$ to $f(A)$ in espectation ?
• What is an upper-bound for $\mathbb P(|f(A+XX^T)-f(A)| > \epsilon)$?