Oops, that’s a typo; it should just be on the RHS.

]]>Regarding “weighted projections”: up to a scale factor (i.e., a single weight , you can view a linear transformation

between Euclidean spaces as an orthogonal projection.

Specifically, if and are subspaces of of dimension and respectively, and is the orthogonal projection onto restricted to , then, the singular values of are the cosines of the principal angles

https://en.wikipedia.org/wiki/Angles_between_flats

If , then these singular values can take any value in $\latex [0,1]$.

So we see that any can be represented as for an appropriate choice of $\latex X,Y$ as subspaces of .

Also, regarding the best orthogonal transformation to represent , it is worth pointing out that you are talking about the orthogonal factor in the polar decomposition

https://en.wikipedia.org/wiki/Polar_decomposition

which is an immediate consequence of the SVD. We can always represent our matrix as a composition of an orthogonal matrix and a positive semidefinte matrix:

,

where and $R= V \Sigma V^T$ and $R’ = U\Sigma U^T$.