Diagonalization of the symmetrized discrete ith right shift operator: an elementary proof

Abstract

In this paper, we consider the symmetric part of the so-called ith right shift operator. We determine its eigenvalues as also the associated eigenvectors in a complete and closed form. The proposed proof is elementary, using only basical skills such as Trigonometry, Arithmetic and Linear algebra. The first section is devoted to the introduction of the tackled problem. Second and third parts contain almost all the “technical” stuff of the proof. Afterwards, we continue with the end of the proof, provide a graphical illustration of the results, as well as an application on the polyhedral “sandwiching” of a special compact of $\mathbb{R}^{n}$ arising in Signal theory.