Let ${\bf X}_{1},\cdots,{\bf X}_{p}$ be $p$ independent Gaussian random variables, in particular, ${\bf X}_{j}\sim N_{n}({\bf 0},\lambda_{j} {\bf I})$, where $\lambda_{j}$s are not all equal, $j=1,\ldots,p$, $n<p$.
Define ${\bf W}=\sum_{j=1}^{p} {\bf X}_{j}{\bf X}_{j}^{\top}$. I am interested in finding the expectation of ${\bf X}_{j}^{\top} {\bf W}^{-1} {\bf X}_{j}$.
Any idea towards solving this would be extremely helpful.
Denote $W_1=\sum_{i=2}^pX_iX_i^T$ and $Y=W_1^{-1/2}X_1$. Then $$Y^TY=\|Y\|^2=X_1^TW_1^{-1}X_1=\mathrm {trace}\, (W_1^{-1}X_1X_1^T)$$ and $$X_1^TW^{-1}X_1=Y^T(I+YY^T)^{-1}Y=Y^T\left(I-\frac{YY^T}{1+\|Y\|^2}\right)Y=\frac{\|Y\|^2}{1+\|Y\|^2}.$$ We have also from the independence $$E(\|Y\|^2)=\mathrm {trace}\,( E(W_1^{-1})E(X_1X_1^T))=\lambda_1\mathrm {trace}\, E(W_1^{-1}).$$Difficult to say more without further hypothesis on the lambdas. Recall that the expectation of the inverse of a Wishart matrix is known, therefore at least $E(\|Y\|^2)$ is computable if $\lambda_2=\ldots=\lambda_p.$