(I'm not sure if this question is appropriate here. If it violates any condition, I will immediately delete it.)

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 $0<\sigma<<1$.

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 $0<\sigma<<1$.

(I'm not sure if this question is appropriate here. If it violates any condition, I will immediately delete it.)

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?

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 $0<\sigma<<1$. Here, $\epsilon = [\epsilon_1 \cdots \epsilon_N]$.

(I'm not sure if this question is appropriate here. If it violates any condition, I will immediately delete it.)

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 $0<\sigma<<1$.