Let $X = [x_1 \cdots x_N] \in \mathbb{R}^{d \times N}$ and $ T= [t_1 \cdots t_N] \in \mathbb{R}^{1 \times N}$. Define $$ E(w) = \text{tr}(T - w^TX)(T - w^TX)^T $$ as least square energy. When defining $$ \epsilon = T- w_{LS}^T X, $$ with $w_{LS}^T$ being the solution that minimizes $E$, can we find a simple bound for $\epsilon$ in terms of $X$ and $T$ without involving inverse operation? And additionally, can I get a bound (needs to be as small as possible) for each $\epsilon_i$ whereas $\epsilon = [\epsilon_1 \cdots \epsilon_N]$?

In addition to the above, regarding the vectors mentioned above as random ones, would there be any way to impose more conditions on the above to derive that $$ \epsilon_i \sim \mathcal{N}(0,I/\sigma^2) $$ with $0<\sigma<<1$.