Let $x_t \in \mathbb{S}^{d-1}$, $\forall t\in \mathbb{N}$ and let $e_1$ be the first cannonical basis vector of $\mathbb{R}^d$, ie, $e_1 = (1,0,\cdots,0)$. Let us form a Gram Matrix $$\sum_{t=1}^T(x_t + e_1)(x_t+e_1)^T $$

If I have a know a complete eigenspace decomposition of the above gram matrix, can I relate it to the unperturbed gram matrix $$\sum_{t=1}^Tx_tx_t^T $$ More specifically if I know $U,\Lambda$ s.t $$U\Lambda U^T = \sum_{t=1}^T(x_t + e_1)(x_t+e_1)^T$$ and if $\tilde{U}, \tilde{\Lambda}$ are s.t. $$\tilde{U}\tilde{\Lambda}\tilde{U}^T = \sum_{t=1}^Tx_tx_t^T$$. Is there any relation that $U \approx \tilde{U}$ and $\Lambda \approx \tilde{\Lambda}$?