expected value of multiplication of random matrices

Question

Suppose $X\in R^{n \times n}$ is a symmetric positive definite random matrix and $A\in R^{n \times n}$ is a symmetric constant matrix. how can I compute the below expression

$$ E[X^TAX] $$

Answer 1

Write $X=[x_1\ x_2\ \cdots\ x_n]$ where $x_i$ are the columns vectors of $X$. Then, the $(i,j)$th element of the matrix $E(X^TAX)$ will be $$E(x_i^TAx_j)=E(Tr(x_i^TAx_j))=E(Tr(Ax_jx_i^T))=Tr(AE(x_j x_i^T))=Tr(AC_{ij})$$ where $C_{ij}$ is the cross correlation of $x_i,x_j$