We assume that $\Sigma_Y$ has $n$ distinct eigenvalues which we order in strictly decreasing order $\lambda_1 > \cdots > \lambda_n$. We equip ourselves with a vector $U_0$ such that $\mathbb{V}\left(U_0^\top Y\right) = \max_{U \in C} \mathbb{V}\left(U^\top Y\right)$, and a vector $U_1 \in C'$ such that $\mathbb{V}\left(U_1^\top Y\right) = \max_{U \in C'} \mathbb{V}\left(U^\top Y\right)$, where $$C' = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \text{ and } U_0^\top U = 0 \right\}.$$ Calculate the covariance of the discrete random variables $U_0^\top Y$ and $U_1^\top Y$ (to simplify notation, one may assume $Y$ is centered, that is, $\mathbb{E}(Y) = 0$).
We assume that $\Sigma_Y$ has $n$ distinct eigenvalues which we order in strictly decreasing order $\lambda_1 > \cdots > \lambda_n$. We equip ourselves with a vector $U_0$ such that $\mathbb{V}\left(U_0^\top Y\right) = \max_{U \in C} \mathbb{V}\left(U^\top Y\right)$, and a vector $U_1 \in C'$ such that $\mathbb{V}\left(U_1^\top Y\right) = \max_{U \in C'} \mathbb{V}\left(U^\top Y\right)$, where
$$C' = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \text{ and } U_0^\top U = 0 \right\}.$$
Calculate the covariance of the discrete random variables $U_0^\top Y$ and $U_1^\top Y$ (to simplify notation, one may assume $Y$ is centered, that is, $\mathbb{E}(Y) = 0$).