Let $X$ be a $p$-vector over $n$ individuals and let $X_1, \dots, X_n$ be the $I$−th observation of dimension $p$. Let $D$ be the diagonal matrix of weights $p_i = \frac1n$.
Let $W = (w_{i,j} = X_i^T M X)_{i,j}$.Suppose that $M = I_p$ and that the PCA of the cloud of X gives p normed principal axes $(u_k)_{k=1,...,p}$ of corresponding eigenvalues $λ_k$ and let $v_k$ denote the associated eigenvectors.
Show that $v_k$ is also an eigenvector of WD.
Consider the vector $f_k \in \mathbb{R}^n$ whose component i is $f_{ik} = \sqrt{p_i}v_{ik}$. Deduce that the matrix WD admits eigenvector $f_k$ with eigenvalue $λ_k$.
Any elpp would be much appreciated. What I have managed to do already is to show that the variance-covariance matrix of $X_i$ is $S=X^TDX$ and that $XSu_k = \lambda_k v_k$.