ON k-CIRCULANT MATRICES INVOLVING THE JACOBSTHAL NUMBERS

Let k be a nonzero complex number. We consider a k-circulant matrix whose first row is (J(1), J(2), ..., J(n)), where J(n) is the nth 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), ..., J(n)(-1)) and obtain the upper and lower bounds for its spectral norm.