On the spectrum of the tridiagonal matrices with two-periodic main diagonal

TL;DR

This paper derives the spectrum and eigenvectors of a class of complex tridiagonal matrices with two-periodic main diagonals, generalizing previous results on Sylvester-Kac matrices and their submatrices.

Contribution

It provides a general method to find spectra and eigenvectors of irreducible complex tridiagonal matrices with two-periodic diagonals, extending known special cases.

Findings

Explicit formulas for spectra and eigenvectors of the matrices.

Generalization of Sylvester-Kac matrix results.

Applicable to a broad class of two-periodic tridiagonal matrices.

Abstract

We find the spectrum and eigenvectors of an arbitrary irreducible complex tridiagonal matrix with two-periodic main diagonal provided that the spectrum and eigenvectors of the matrix with the same sub- and superdiagonals and zero main diagonal is known. Our result substantially generalises some recent results on the Sylvester-Kac matrix and its certain main principal submatrices.