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##### 1936-1944
On $k$-circulant matrices involving the Jacobsthal numbers
Volume 60, no. 2 (2019), pp. 431–442

### Abstract

Let $k$ be a nonzero complex number. We consider a $k$-circulant matrix whose first row is $(J_{1},J_{2},\dots,J_{n})$, where $J_{n}$ is the $n^\text{th}$ Jacobsthal number, and obtain the formulae for the eigenvalues of such matrix improving the formula which can be obtained from the result of Y. Yazlik and N. Taskara [J. Inequal. Appl. 2013, 2013:394, Theorem 7]. The obtained formulae for the eigenvalues of a $k$-circulant matrix involving the Jacobsthal numbers show that the result of Z. Jiang, J. Li, and N. Shen [WSEAS Trans. Math. 12 (2013), no. 3, 341–351, Theorem 10] is not always applicable. The Euclidean norm of such matrix is determined. We also consider a $k$-circulant matrix whose first row is $(J_{1}^{-1},J_{2}^{-1},\dots,J_{n}^{-1})$ and obtain the upper and lower bounds for its spectral norm.