Evaluate $tr(ADAD)$ where $D$ is a diagonal matrix with iid entries from $N(0,1)$ on diagonal and $A$ is a deterministic $n \times n$ matrix

Question

Let $D$ be a random $n \times n$ matrix with iid entries from $N(0,1)$ on the diagonal and let $A$ be a symmetric deterministic $n \times n$ matrix.

Question. What does $tr(ADAD)$ evaluate to ?

Answer 1

Write $A = [a_{ij}]_{1\leq i,j\leq n}$ and $D = \operatorname{diag}(X_1,\ldots,X_n)$. Then

$$ \operatorname{tr}(ADAD) = \sum_{i,j,k,l} a_{ij}(X_j \delta_{jk}) a_{kl} (X_l \delta_{li}) = \sum_{i,j} a_{ij}^2 X_i X_j. $$

So, if we write $B = [a_{ij}^2]_{1\leq i,j \leq n}$ and $X = \begin{pmatrix} X_1 & X_2 & \ldots & X_n \end{pmatrix}^{\mathsf{T}}$, then

$$ \operatorname{tr}(ADAD) = X^{\mathsf{T}} B X. $$

This has a generalized chi-square distribution.