Block diagonalization of ( p, q )-tridiagonal matrices

Abstract In this article, we study the block diagonalization of ( p , q ) \left(p,q) -tridiagonal matrices and derive closed-form expressions for the number and structure of diagonal blocks as functions of the parameters p p , q q , and n n . This reduction enables efficient computation of eigenvalues and eigenvectors by decomposing the matrix into smaller subproblems. We extend the method to more general ( P , Q ) \left({\mathcal{P}},{\mathcal{Q}}) -tridiagonal matrices, where P {\mathcal{P}} and Q {\mathcal{Q}} are sets of positive integers, covering general banded structures. We also examine special cases such as bidiagonal and triangular block reductions along with supporting algorithms and numerical examples.