On k-circulant matrices involving the Pell–Lucas (and the modified Pell) numbers

Abstract

Let k be a nonzero complex number. In this paper, we consider a k-circulant matrix whose first row is $$(Q_{1},Q_{2},\ldots ,Q_{n})$$ , where $$Q_{n}$$ is the nth Pell–Lucas number. The formulas for the eigenvalues of such matrix are obtained. Namely, the result which can be obtained from the result of Theorem 7. (Yazlik and Taskara, J Inequal Appl 2013:394, 2013) is improved. The obtained formulas for the eigenvalues of a k-circulant matrix involving the Pell–Lucas numbers show that the result of Theorem 8. (Jing, Li and Shen, WSEAS Trans Math 12(3):341-351, 2013) (i.e. Theorem 8. (Yazlik and Taskara 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 $$(Q_{1}^{-1},Q_{2}^{-1},\ldots ,Q_{n}^{-1})$$ are also investigated. The obtained results are illustrated by examples. As a consequence of the previous results, the eigenvalues, the determinant, the Euclidean norm of a k-circulant matrix whose first row is $$(q_{1},q_{2},\ldots ,q_{n})$$ , where $$q_{n}$$ is the nth modified Pell number, are presented. Also, the upper and lower bounds for the spectral norm of a k-circulant matrix whose first row is $$(q_{1}^{-1},q_{2}^{-1},\ldots ,q_{n}^{-1})$$ are given