# Orthogonality leading to a simpler optimization problem (procrustes, Stiefel manifold)

I have seen somewhere doing the following "trick" to simplify an optimization problem. I would like to understand the logic behind it.

Take two matrices $$A$$ and $$B$$ of dimension $$N \times N$$ and suppose that matrix $$B$$ is function of a parameter $$\theta$$, so we have $$B(\theta)$$. I have seen doing the following:

\begin{align} \min_{\theta} \|A - B(\theta)B(\theta)^\top\|_F^2 \rightarrow \min_{\theta, U} \|A^{1/2} - B(\theta)U\|_F^2 \end{align}

where $$A^{1/2}$$ is a square root of $$A$$ and $$U$$ is an orthogonal matrix, i.e. $$UU^\top=U^\top U=I$$. This would lead to an optimization problem which is different from the initial one but easier in the variable $$\theta$$, since reduces its order (for instance from quadratic to linear), at the expenses of also estimating the matrix $$U$$. Computing $$U$$ can be done through the procrustes method. Sometimes such a problem is said to be "over the Stiefel manifold".

Can you elucidate me how the two problems are related? Thanks.