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.
Similar content being viewed by others
References
Alkire, B., Vanderberghe, L.: Convex optimization problems involving finite autocorrelation sequences. Math. Program. 93, 331–359 (2002)
Fuentes, M.: Analyse convexe et optimisation sur le cône des vecteurs à composantes autocorrelées. Technical report, Université Paul Sabatier, Toulouse. Available at: http://www.mip.ups-tlse.fr/publis/files/05.58.htm (2005)
Strang, G., Gorsich, M., Genton, M.: Eigenstructures of spatial design matrices. J. Multivar. Anal. 80, 138–165 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fuentes, M. Diagonalization of the symmetrized discrete ith right shift operator: an elementary proof. Numer Algor 44, 29–43 (2007). https://doi.org/10.1007/s11075-007-9076-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-007-9076-4