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 $tr(XAA^TX^T)+\sum_i\alpha_i(x_i^Tx_i-1)+\sum_i\sum_{j\neq i}\beta_{ij} q_i^Tq_j$?
