Essentially, I'm wondering if there's a list of the systems where there's a known analytic eigendecomposition (eigenvalues and eigenvectors) for $Ax = \lambda x$.

For example, tridiagonal matrices have elementary expressions for its eigenvalues and eigenvectors. Block multi-diagonal matrices also have analytic expressions for their eigenvalues, see Section 4 of http://repository.uwyo.edu/cgi/viewcontent.cgi?article=1600&context=ela. However, there don't appear to be simple analytic forms for their eigenvectors.

I'm hoping for solutions in the form of elementary functions, but analytic functional forms that are easy to evaluate (even numerically) is also fine.

The reason I ask is that Cholesky factorization is a thing, and permutations of our matrix $A$ could yield simpler forms that may have analytic expressions for its eigenvalues/eigenvectors. I want to see if any of this could be useful to my work in condensed matter physics.