Analytic form of eigenpairs for a special matrix

[robertsj]

Hi all,

I have a physically-motivated algorithm for which I'm trying to flesh out some basic properties analytically. In one case, I end up with a matrix of the following form:

$$\left [\begin{array}{ccccccccc} 0 & & & & & & & & \\ & 0 & R & T & & & & & \\ T & R & 0 & & & & & & \\ & & & 0 & R & T & & & \\ & & T & R & 0 & & & & \\ & & & & & \ddots & & & \\ & & & & & & 0 & R & T \\ & & & & & T & R & 0 & \\ & & & & & & & & 0 \\ \end{array} \right ]$$

I can compute the the fundamental mode analytically based on the physics of the problem, but I haven't been able to generate higher order modes. I'm most interested in the eigenvalues. Any suggestions? Has someone done this?

 

[fresh_42]

It should be relatively easy to calculate the determinant for the eigenvalue polynomial by the closed Laplace formula: https://en.wikipedia.org/wiki/Determinant#Laplace's_formula_and_the_adjugate_matrix and achieve a recursion quickly.