I am having trouble understanding how to get the weighted generalized inverse of a matrix. Let me start from the beginning.
Suppose $$a=Xb$$
Where $a$ is a vector with m elements, $b$ is a vector of n elements and $X$ is a matrix with mxn elements. Solving for $b$ we get $$b={X}^{+}a$$
Where ${X}^{+}$ is the Moore-Penrose inverse defined as
$$X^+ = X^T (XX^T)^{-1}$$
Where $X^T$ is the transpose of $X$. The weighted generalized inverse is given by
$$X_W= {W}^{-1}X^T (X{W}^{-1}X^T)^{-1}$$
Making the weighted solution of $b$ $$b_W= {W}^{-1}X^T (X{W}^{-1}X^T)^{-1}a$$
The proofs that I have found start from places that I don't understand how they end up there to begin with. I probably don't know something relatively basic, but can someone tell me where I can find the proof without taking any assumption for granted?
Any help would be appreciated.
