Suppose that I have an $nk \times nk$ matrix of the form $$ T_n = \left[\begin{array}{cccccc} A&B&&&&\\ B^T&A&B&&&\\ &B^T&A&B&&\\ &&\ddots&\ddots&\ddots&\\ &&&B^T&A&B\\ &&&&B^T&A\\ \end{array} \right] $$ where $A$ and $B$ are $k \times k$ real matrices (with $A$ symmetric).
This is a symmetric block-tridiagonal matrix.
I would like an expression for the characteristic polynomial of $T_n$, or even a recurrence to compute the characteristic polynomial of $T_n$.
If I had the slightly different matrix $$ T'_n = \left[\begin{array}{cccccc} A&B&&&&B^T\\ B^T&A&B&&&\\ &B^T&A&B&&\\ &&\ddots&\ddots&\ddots&\\ &&&B^T&A&B\\ B&&&&B^T&A\\ \end{array} \right] $$ which is a block-circulant matrix, then there is a nice expression to be found in Denis Serre's answer to the question:
One question on block-circulant matrices
which can be used to express the characteristic polynomial as a product of determinants of various matrices.
There are various papers about block tridiagonal matrices, in particular one by Molinari (https://arxiv.org/abs/0712.0681), that seems initially promising, but this requires the off-diagonal blocks to be invertible (mine are not invertible).
There is also the question Determinant of block tridiagonal matrices that asks the same thing (essentially) for a particular instance of a matrix of this form where $A = J_k$ and $B = I_k$ ($J_k$ is the all-ones matrix of order $k \times k$). My blocks do not commute so this answer does not work for me.
Any pointers or references gratefully received.
