Realization: the binary multiplicative structure can probably be recovered fairly well from the binary additive structure + unary eigendecomposition?
Let’s say you’ve got three subspaces X, Y and Z (represented as projection matrices). Imagine that one prompt uses dimensions D1=X+Y, and another prompt uses dimensions D2=Y+Z. If we take the difference, we get X−Z. Notably, the positive eigenvalues correspond to X, and the negative eigenvalues correspond to Z.
Define f(P) to yield the part of P with positive eigenvalues (which I suppose for projection matrices has a closed form of P+P22, but the point is it’s unary and therefore nicer to deal with mathematically). You get D1∧¬D2=f(D1−D2), and you get D1∧D2=D1−f(D1−D2).
Realization: the binary multiplicative structure can probably be recovered fairly well from the binary additive structure + unary eigendecomposition?
Let’s say you’ve got three subspaces X, Y and Z (represented as projection matrices). Imagine that one prompt uses dimensions D1=X+Y, and another prompt uses dimensions D2=Y+Z. If we take the difference, we get X−Z. Notably, the positive eigenvalues correspond to X, and the negative eigenvalues correspond to Z.
Define f(P) to yield the part of P with positive eigenvalues (which I suppose for projection matrices has a closed form of P+P22, but the point is it’s unary and therefore nicer to deal with mathematically). You get D1∧¬D2=f(D1−D2), and you get D1∧D2=D1−f(D1−D2).
Maybe I just need to do epic layers of eigendecomposition...