How to show that the following summation is true for matrix multiplication?

Question

I have following matrices $$\mathbf{b}\in R^n,~\mathbf{W}\in R^{n\times n},~ A\in R^{n\times m}.$$ Where $\mathbf{W}$ is a diagonal matrix and let us assume that the $i$-th row of $\mathbf{A}$ is written as $\mathbf{a}_i$ and the $i,i$-th entry of $\mathbf{W}$ is $w_i$. In this how to show that the following equality is true $$\mathbf{b^TWA(A^TWA)^{-1}A^TWb}=\sum_{i=1}^{n}w_i^2b_i^2\mathbf{a}_i^T\left(\sum_{j=1}^nw_j\mathbf{a}_j\mathbf{a}_j^T\right)^{-1}\mathbf{a}_i$$ where $b_i$ is the $i$-th entry of $\mathbf{b}$. Thanks in advance.

Answer 1

Isn't simply $$\mathbf b^T\mathbf W\mathbf A(\mathbf A^T\mathbf W\mathbf A)^{-1}\mathbf A^T\mathbf W\mathbf b =\mathbf b^T\mathbf W\mathbf A \mathbf A^{-1}\mathbf W^{-1}(\mathbf A^T)^{-1}\mathbf A^T\mathbf W\mathbf b=\mathbf b^T\mathbf W\mathbf b $$ ?