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?

Apparently, one possible answer is $$ E[N^2] = 3\ \text{Tr} (A A' A A') + 2 \sum_{\text{all } A_i 2\times 2 \text{ submatrices}} \text{Det} (A_i)^2. $$ Does this sound familiar to anyone ? 

