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.

In condensed matter physics, we often come across matrices that are multi-diagonal or banded. For example, I may have a matrix with three tridiagonal bands, or a tridiagonal band and two/four bidiagonal bands. Some groups are also working on essentially banded matrices with a pentadiagonal band and then two/four quad-diagonal bands.

Essentially, I'm wondering if there's a list of the systems where there's a known analytic eigendecomposition (eigenvalues and eigenvectors) for $Ax = \lambda x$, at least for the types of cases I've listed above. If so, then I'm also interested in learning more about what systems can reduce down to matrices of the above forms through permutations or similarity transforms.

In particular, in Numerical Analysis of Differential Equations by Iserles, he states in Chapter 11 that some matrices have 'perfect' Cholesky factorizations, wherein you have banded matrices with no fill-in. No-fill-in one-band matrices come about in one-dimensional condensed matter problems, but not 2D or 3D problems.

I am interested in investigating what kinds of systems can Cholesky-factorize into matrices that have analytic eigendecompositions. I'm not sure if there are resources for this, which is why I'm willing to investigate it myself - but I haven't been able to find lists of matrices with known eigendecompositions.

To start the list, tridiagonal matrices have elementary expressions for its eigenvalues and eigenvectors. See, for example, works related to Molinari et al. 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.