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Results in Mathematics

, 74:200 | Cite as

On k-Circulant Matrices Involving the Pell Numbers

  • Biljana RadičićEmail author
Article
  • 60 Downloads

Abstract

Let k be a nonzero complex number. In this paper, we consider a k-circulant matrix whose first row is \((P_{1},P_{2},\dots ,P_{n})\), where \(P_{n}\) 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 \((P_{1}^{-1},P_{2}^{-1},\dots ,P_{n}^{-1})\) are also investigated. The obtained results are illustrated by examples.

Keywords

k-circulant matrix Pell numbers eigenvalues Euclidean norm spectral norm 

Mathematics Subject Classification

Primary: 15B05 Secondary: 11B39 15A18 15A60 

Notes

Acknowledgements

We would like to thank the anonymous reviewer for his (or her) careful reading of our manuscript and his (or her) suggestions.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Singidunum UniversityBelgradeSerbia

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