Assume $X\sim \mathcal{N}(0,I)$, $X\in \mathbb{R}^{Nx2}$ and $V\in \mathbb{R}^{2x2}$ is a matrix of constants
I'm trying to find out how to evaluate: $$\mathbb{E}[X V^TVX^T]$$
When the $X's$ are on the inside this is easy: $$\mathbb{E}[VXX^TV^T] = V\mathbb{E}[X^TX]V^T = VIV^T$$
But I cant wrap my head around when the $X$'s are on the outside... Any hints?
Hint: The matrix $XV^TVX^T$ is an $n\times n$ matrix of random variables, so its expectation is defined elementwise. To find the expectation of the $(i,j)$ element of this matrix, go back to the definition of matrix multiplication: $$ (XV^TVX^T)_{i,j}=\sum_k\sum_l X_{i,k}(V^TV)_{k,l}(X^T)_{l,j} =\sum_k\sum_l (V^TV)_{k,l} X_{i,k}X_{j,l}\tag1 $$ and then apply expectation to (1), which is a linear combination of random variables: $$ {\mathbb E}[(XV^TVX^T)_{i,j}]=\sum_k\sum_l (V^TV)_{k,l}{\mathbb E}[ X_{i,k}X_{j,l}]\tag2 $$ Now compute ${\mathbb E}[ X_{i,k}X_{j,l}]$ using what you know about the joint distribution of the elements of $X$.
It's not clear if the rows or the columns of $X$ follow a ${\mathcal N}(0,I)$ distribution, so I'll leave it to you to finish the calculation. Note that ${\mathbb E}[ X_{i,k}X_{j,l}]$ will be zero for certain combinations of $i,j,k,l$, which causes simplification in (2).