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Consider the problem:

$\min_X ||XAX^T||_F$

s.t. $X^TX=I$

If A and X are real matrices, the lagrangian will be $tr(XAX^TXA^TX^T)+\sum_i\alpha_i(x_i^Tx_i-1)+\sum_i\sum_{j\neq i}\beta_{ij} q_i^Tq_j$ where $\alpha_i$ and $\beta_{ij}$ are Lagrangian multipliers.

Can I apply $X^TX=I$ here such that the Lagrangian becomes $tr(XAA^TX^T)+\sum_i\alpha_i(x_i^Tx_i-1)+\sum_i\sum_{j\neq i}\beta_{ij} q_i^Tq_j$?

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This question would be more appropriate on math.SE –  Suresh Venkat Dec 28 '12 at 6:55
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