What real symmetric matrices of this type can Mathematica find symbolic eigenvalues for?

I'm working on a problem calculating symbolic eigenvalues of matrices that always have a very simple form: they are real and symmetric and usually sparse. They have two distinct symbolic parameters. (See below for an example) The matrices can be 2x2 or in principle as large as you like, but in practice up to maybe 20x20. I want the symbolic eigenvalues of these matrices and of course sometimes the characteristic polynomials are factorable and Mathematica can calculate symbolic answers, and sometimes Mathematica (or anyone!) can't. Whether or not symbolic answers can be provided does not depend on matrix size but rather how the off-diagonal elements are located.

Is there some way to know what kinds of matrices in these sorts of problems will lead to simple symbolic eigenvalues, and which ones won't?

I'm not sure how much of this question is about algebra in general, versus what Mathematica's symbolic manipulation routines are capable of.

As an example, here's what one of those matrices looks like:

matrixA={
{α, β, 0, 0, 0, 0, β, 0, 0, β},
{β, α, β, 0, 0, 0, 0, 0, 0, 0},
{0, β, α, β, 0, 0, 0, 0, 0, 0},
{0, 0, β, α, β, 0, 0, 0, 0, 0},
{0, 0, 0, β, α, β, 0, 0, 0, 0},
{0, 0, 0, 0, β, α, β, 0, 0, 0},
{β, 0, 0, 0, 0, β, α, β, 0, 0},
{0, 0, 0, 0, 0, 0, β, α, β, 0},
{0, 0, 0, 0, 0, 0, 0, β, α, β},
{β, 0, 0, 0, 0, 0, 0, 0, β, α}
}


For reference, I asked a more elementary question about this a few years ago and some of the replies were really helpful - link to that question here: Trying to simplify Root expressions from the output of Eigenvalues

• What is your criterion for a symbolic eigenvalues to be called "simple"? The zeroes of any polynomial can be expressed by Mathematica in terms of Root functions, as noted an answer to your earlier question. Would a cubic factor be considerer simple? Or, a quartic factor? Or, ... Commented Mar 14, 2017 at 4:16
• Your example has: a) a constant diagonal b) parameter α only appearing on the diagonal. Is this a common feature of your matrices? If so, your problem can simplified to dependence on a single parameter, or even parameter independence by using simple transforms. Commented Mar 15, 2017 at 20:09
• @bbgodfrey - My criterion for 'simple' is a closed form algebraic expression. Root functions would not be that. Any algebraic expression would be fine. Commented Mar 17, 2017 at 17:51
• @mikado - Yes those are both essential factors for all the matrices I deal with. I'm aware I could rescale that matrix to one single parameter (I don't think independent of all parameter though!) and numerically calculate eigenvalues. I'm looking for an answer to the question as posed because it seems like an interesting algebra/Mathematica question that would arise naturally because the way I showed the matrices is the way they would naturally be first written down by people working these problems (Huckel molecular orbital theory, if it matters) Commented Mar 17, 2017 at 17:54
• You might consider posting this question in Mathematics. Commented Mar 17, 2017 at 19:22