How do I compute the following expectation for a matrix random variable?

Question

I've successfully compute the expectation for the quadratic form of a random vector. But I've stumbled upon this matrix form. Let $X$ be a $n \times m$ random matrix and $A$ a $n \times n$ square constant matrix. How do I compute $$ \mathbb{E}\left[X^{T}A X\right] ?$$ I know that $$\operatorname{Var}\left[ AX\right] = A\operatorname{Var}\left(X\right)A^{T}.$$ But I'm not sure how this relates to my problem.

Answer 1

Each entry in the product $X^TAX$ is a quadratic form in the entries of $X$. Recall the expectation value of a matrix $M$ is defined as

$$E[M] : = \pmatrix{E[M_{11}] & E[M_{12}]& \cdots \\ E[M_{21}] & E[M_{22}] & \cdots \\ \vdots & \vdots & \ddots }$$

Since you know how to compute expectation values of quadratic forms, this should be a piece of cake.