# LDA, Significance of orthonormality- Trace Ratio Maximization

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..

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Could you give us a better explanation of the context of your question? – zeferino Mar 26 '12 at 12:38
The reference to "different views" appears to be a solicitation to a discussion. That would be appropriate in chat, but not as a question. Could you perhaps reformulate the question in a form acceptable to this site? – whuber Mar 26 '12 at 18:43
Changed/Reformulated as required. – Praneeth Vepakomma Mar 26 '12 at 18:53