On k-Circulant Matrices Involving the Pell Numbers

Let k be a nonzero complex number. In this paper, we consider a k-circulant matrix whose first row is (P1,P2,⋯,Pn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_{1},P_{2},\dots ,P_{n})$$\end{document}, where Pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{n}$$\end{document} is the nth Pell number, and obtain the formulae for the eigenvalues of such matrix improving the result which can be obtained from the result of Theorem 7 (Yazlik and Taskara in J Inequal Appl 2013:394, 2013). The obtained formulae for the eigenvalues of a k-circulant matrix involving the Pell numbers show that the result of Theorem 6 (Jiang et al. in WSEAS Trans Math 12(3):341–351, 2013) [i.e. Theorem 8 (Yazlik and Taskara in J Inequal Appl 2013:394, 2013)] is not always applicable. The Euclidean norm of such matrix is determined. The upper and lower bounds for the spectral norm of a k-circulant matrix whose first row is (P1-1,P2-1,⋯,Pn-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_{1}^{-1},P_{2}^{-1},\dots ,P_{n}^{-1})$$\end{document} are also investigated. The obtained results are illustrated by examples.