Let $A$ be an $n \times d$ matrix ($n > d$) with iid standard Gaussian entries. Let $\lambda_1 \ge \lambda_2 \ge \dotsm \lambda_d > 0$ be the non-zero singular values of $A$. What is known about the quantity $Q = \sum_{j=1}^d \lambda_j^4$?
I am particularly interested in the asymptotic behavior as $n,d \to \infty$ while $n/d \to r \in (0,\infty)$, as well as the expectation of $Q$.