Linear Discriminant - Least Squares Classification Bishop 4.1.3
As Bishop points out throughout that section, least squares is ill-equipped for this problem, so maybe we shouldn't spend too much time understanding it.  On the other hand, clearing up misconceptions here may help elsewhere.


We would like for (i.e., least squares strives for) $\operatorname{val}(\mathcal{C}_2)$ to be close to 1 and the others close to 0.  But the values could very well be negative, positive, greater than 1, for any of the classes (the correct ones or not)!  Our final classification though is dealt with as Section 4.1.2 explains (just after equation 4.9): the class with largest $y$-value wins.
Yes, here $\operatorname{Tr}$ means the trace.  In "least squares," we're minimizing the mean squared error.  It happens that the sum of squared errors can be written nicely as the trace of these matrices, but the incidental entries of the matrix off the diagonal don't concern us here.