Linear Discriminant - Least Squares Classification Bishop 4.1.3

Pls. refer section 4.1.3 in Pattern Recognition – Bishop: "Least squares for Classification":

In a 2 class Linear Discriminat system, we classified vector $mathbf{x}$ as $mathcal{C}_1$ if y($bf{x}$)>0, and $mathcal{C}_2$ otherwise.
Generalizing in section 4.1.3, we define $mathcal{K}$ linear discriminant equations – one for each class:

$y_{k}(bf{x}) = bf{w_k^Tx} + mathit{w_{k0}} tag {4.13}$

adding a leading 1 to vector $bf{x}$ yields $tilde{mathbf{x}}$.

And the Linear Discriminant function for $mathcal{K}$ class is given by:
$bf y(x) = widetilde{W}^{T}tilde{x}$. The author progresses and presents sum of squares Error function as:

$E_D(widetilde{W}) = frac{1}{2}Tr{(tilde{X}widetilde{W} – T)^T(tilde{X}widetilde{W} – T)} tag {4.15}$

My doubts are related to above equation 4.15.

Consider a 3-class system with only one observation, $bf{x}in mathcal{C_2}$, my understanding:

1. Pls. refer $bf{Y}$ in upper half of diagram. Will only $val(mathcal{C_2})$ be positive: $mathbf{x} in mathcal{C_2}$,
   $y_{2}(bf{x})$ > $it{threshold}(mathcal{C_2})$. Is the value,
   $val(mathcal{C_k})$, negative for other classes’ Discriminant
   functions? If not, could you briefly explain the reason?
2. The error matrix, $bf{E}$ is 1×3 matrix. $bf{E}^{T}E$ will be a 3×3 matrix, with diagonal elements representing squared(Error) for a
   class. Does $Tr$ in 4.15 stand for $trace$ – sum of diagonal elements?
   If so, why do we ignore off diagonal error values/ why don’t they
   matter?

P.S.: If my understanding is wrong/ grossly wrong, I’ll appreciate if you point out the same.