Abstract
Let k be a nonzero complex number. We consider a k-circulant matrix whose first row is (J1, J2, . . . , Jn), where Jn is the n 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−1 2 , . . . , J−1 n ) and obtain the upper and lower bounds for its spectral norm
Highlights
Throughout this paper k is a nonzero complex number.Definition 1.1
We shall improve the result in relation to the eigenvalues of (1.2) which can be obtained from the formula for the eigenvalues of a k-circulant matrix with the generalized r-Horadam numbers {Hr,n} (the numbers defined as follows: Hr,n+2 = f (r)Hr,n+1 + g(r)Hr,n, n ≥ 0, where r ∈ R+, Hr,0 = a, Hr,1 = b, a, b ∈ R, and f 2(r) + 4g(r) > 0), presented in [26], because the authors did not consider the case when the denominator is equal to zero
Before we present our main results, let us recall that the Jacobsthal numbers {Jn} satisfy the following recurrence relation: Jn = Jn−1 + 2Jn−2, n ≥ 2, with initial conditions J0 = 0 and J1 = 1
Summary
Throughout this paper k is a nonzero complex number. Definition 1.1. We shall improve the result in relation to the eigenvalues of (1.2) which can be obtained from the formula for the eigenvalues of a k-circulant matrix with the generalized r-Horadam numbers {Hr,n} (the numbers defined as follows: Hr,n+2 = f (r)Hr,n+1 + g(r)Hr,n, n ≥ 0, where r ∈ R+, Hr,0 = a, Hr,1 = b, a, b ∈ R, and f 2(r) + 4g(r) > 0), presented in [26], because the authors did not consider the case when the denominator is equal to zero. Before we present our main results, let us recall that the Jacobsthal numbers {Jn} satisfy the following recurrence relation: Jn = Jn−1 + 2Jn−2, n ≥ 2, with initial conditions J0 = 0 and J1 = 1.
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