Let $X$ be a gaussian vector in dimension $n$, with $0$ mean and covariance identity. Let $A$ be a square matrix of size $n$, and $Y = A X$. Let $N$ be the square of $Y$ euclidean norm: $N = \sum Y_i^2$. One computes easily the mean of $N$: $E[N] = \text{Tr}(A A')$. But what about
its variance ?