Expectancy of $X^TAX$ when $X$ is a matrix

Question

To simplify, let's suppose that $E(X)=0$ and $A$ an idempotent and symmetrical matrix; that is $A^TA=A$.

If $X$ is a vector, It is shown that $$E(X^TAX) = E\left(Tr(X^TAX)\right) = E\left(Tr(AXX^T)\right) = Tr\left(A E(XX^T)\right)$$

But, how can one compute $E(X^TAX)$ when $X$ is a matrix? In fact, $X^TAX \ne Tr(X^TAX)$

Answer 1

It works basically in the same way.

Let $X_i$ denotes the $i$-th column of $X$. Then, the $(i,j)$-th component of the matrix $E(X^T A X)$ is given by $$ E(X^T A X)_{i,j} = E(X_i^T A X_j) = E(Tr(X_i^T A X_j)) = Tr(A E(X_j X_i^T)). $$