The objective of fisher linear discriminant analysis can be formulated as maximizing $\frac{Tr[X^TAX]}{Tr[X^TBX]}$ over $X$ where $A$ and $B$ are positive semi-definite with orthonormality constraints over X.
What is the significance of the orthonormality constraint in here?
I am also curious if reformulating this objective as $Tr[X^TAX]-\alpha Tr[X^TBX]$ is justified? $\alpha$ belongs to positive reals and the objective is a function of $X$ and $\alpha$ in this case.
Was reading section II in : http://www.eecs.berkeley.edu/~jiayq/papers/tnn_traceratio.pdf and was wondering about the orthonormality constraint. I believe that orthonormality constraints in some trace minimization problems prevent a trivial solution, where the entries in X can all go to zero. The objective for Fisher's LDA seems to be having the orthonormality constraint for quite a different reason, and am not sure about it. Has it got do to something with a unique solution over rotation matrices? I don't believe that to be true either..
