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 $R’ = U\Sigma U^T$.

In particular, take the subobject classifier for (directed multi)graphs, aka quivers; is there a presentation of this Lawvere theory with nullary, unary, and binary function symbols like the one in your post? What are the models? Since discrete graphs are the same as sets, it seems like the algebras would be special Boolean algebras.

]]>See http://mathoverflow.net/questions/168888/who-invented-diagrammatic-algebra

The thesis (in German) can be download here:

]]>