I believe "speed up" in this sense is really akin to how Cramer's rule lets you write down a relatively neat formula for the solution of a linear system, not th

I believe "speed up" in this sense is really akin to how Cramer's rule lets you write down a relatively neat formula for the solution of a linear system, not that it would actually help with performing the calculation numerically. Computing a full set of eigenvalues of a numerical matrix is pretty slow and the eigenvectors almost come for free; it seems like a very computationally backwards way to get n+1 full sets of eigenvalues only to get slightly incomplete data about the values of the eigenvectors. I don't know anything about the physics of it, but I doubt that the researchers had numerical matrices that this formula helped with; I expect it was all done symbolically instead.