We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. We assume $r < n$ where $r$ is the rank of $\Sigma_Y$. We denote by $d = \dim \ker \Sigma_Y$ and we consider an orthonormal basis $(V_1, \ldots, V_d)$ of $\ker \Sigma_Y$.
Prove that
$$\forall j \in \llbracket 1, d \rrbracket, \quad \mathbb{V}\left(V_j^\top(Y - \mathbb{E}(Y))\right) = 0.$$