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For positive semi-definite matrices $A,B$, how can I find an $X$ that minimizes $\text{Trace}(AX^TBX$) under the constraint:

that $X$ is orthogonal.

All the matrices have real entries and $A,B$ are square while $X$ is rectangular. Thanks.

This is what I have:

Define $B=F^{T}F$. Define $Y=FX$. You get the above problem as \begin{align} \min_{Y}~ \text{trace}(AY^{T}Y) \end{align}

But now I want $X^*$ that minimizes the original problem. This is what is confusing me!

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    $\begingroup$ If you observe that $A = G^\top G$, and then apply the cyclic invariance property of the trace, you get a minimization of trace$(Z^T Z) = \|Z \|_F^2$ where $Z = F X G^\top$. That should simplify things somewhat, especially since $F$ and $G$ are products of orthogonal matrices with a diagonal matrix. $\endgroup$ – Suresh Venkat Nov 19 '12 at 16:22

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