The following problem arose while studying a problem involving variance equality of two estimators.
Let us consider a set of vectors with positive elememts: $x_i=(1,x_{1i})'$, a set of real numbers: $f_i>0$ and a set of positive integers: $n_i$, $i=1,\ldots,k.$
Now define few matrices: $X'=[x_1,\ldots,x_k],$
$$\Omega= \begin{bmatrix} \frac{1}{n_1f^2_1} & \ldots &0\\ \ldots & \ldots&\ldots\\0 &\ldots& \frac{1}{n_kf^2_k}\end{bmatrix},$$ a diaginal matrix,
and two positive definite matrices: $D_0=\sum_{i=1}^kn_ix_ix'_i$ and $D_1=\sum_{i=1}^kf_in_ix_ix'_i$
I like to check the relation between $D_1D_0^{-1}D_1$ and $X'\Omega^{-1}X$. That is whether:
$$D_1D_0^{-1}D_1=X'\Omega^{-1}X ?$$
I tried to solve this by deriving the above product of the matrices term by term. But stucked, in finding relationship between the expanded term. I guess matrix algebra may be useful here. Computation of the above two matrices, for some example gives indication that the two sides are equal.
Any helpful suggestion highly appreciable.