A proof about generalized eigenvalue problem

Question

I'm considering the following model.
Given $x_i\in\mathbb{R}^n$, $i\in \{1,2,\ldots,N\}$, where $n\leq N$. $X=[x_1,x_2,\ldots,x_N]$ and a full-rank $W\in \mathbb{R}^{N\times N}$
$P=[p_1,p_2,\ldots,p_d]$, where $d<n$ and $p_i$ is the generalized eigenvector vector corresponding to the $i$-th smallest eigenvalues of the following problem $$XWX^Tp=\lambda XX^Tp.$$ I want to show that $$x_i=P(P^TP)^{-1}P^Tx_i,\forall\ i\in{1,2,\ldots,N}.$$ I don't know how to start the proof.
Any help would be appreciated!

Answer 1

This is not true. If $W=I_N$ and $\lambda=1$, the equation $XWX^Tp=\lambda XX^Tp$ is always satisfied regardless of the values of $X$ and $p$. Now, if $n>d=1$, take any $p\ne0$. Then $p(p^Tp)^{-1}p^T$ is singular. Hence there exists some vector $x$ such that $x\ne p(p^Tp)^{-1}p^Tx$. Duplicate $x$ to form a matrix $X$. Now we get a counterexample.