> That would be less interesting, since it's the method taught in every first course of linear algebra, right? But it would be interesting to mention that that

> That would be less interesting, since it's the method taught in every first course of linear algebra, right? But it would be interesting to mention that that's what this method is doing, just using that, in the distinct-eigenvalues case, A - \beta I maps onto the \alpha-eigenspace, so you can simply apply it instead of explicitly finding eigenvectors. In fact, every time I teach linear algebra, one of my students discovers this 2×2 trick for finding eigenvectors, though usually accidentally thinking that they are finding \beta-eigenvectors, rather than \alpha-eigenvectors, by computing the image of A - \beta I instead of its kernel.