Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal–Lucas Numbers

Abstract

Let n ≥ 3 and Jn := circ(J1, J2, . . . , Jn) and ‫ג‬n := circ(j0, j1, . . . , jn−1) be the n × n circulant matrices, associated with the nth Jacobsthal number Jn and the nth Jacobsthal-Lucas number jn, respectively. The determinants of Jn and ‫ג‬n are obtained in terms of the Jacobsthal and Jacobsthal-Lucas numbers. These imply that Jn and ‫ג‬n are invertible. We also derive the inverses of Jn and ‫ג‬n), i.e., where J k and j k are the kth Jacobsthal and Jacobsthal-Lucas numbers, respectively, with the recurrence relations J k = J k−1 + 2J k−2 , j k = j k−1 + 2j k−2 , and the initial conditions J 0 = 0, J 1 = 1, j 0 = 2, and j 1 = 1 (k ≥ 2). Let α and β be

On the determinants of some kinds of circulant-type matrices with generalized number sequences

“…For example, Shen et al gave determinants and inverses of circulant matrices with Fibonacci and Lucas numbers [1]. Bozkurt and Tam considered determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers [2]. Moreover, Bozkurt obtained determinants and inverses of circulant matrices with a generalized number sequences [3].…”

Determinants and inverses of circulant matrices with complex Fibonacci numbers

“…On this topic, Bozkurt and Tam [2] obtained analogues of the results of the paper [9] for circulant matrices associated with Jacobsthal and Jacobsthal-Lucas numbers. Furthermore, Bozkurt and Tam [3] gave a generalization of aforementioned determinant formulae and results on invertibility of these particular circulant matrices.…”

Determinant and inverse of a Gaussian Fibonacci skew-Hermitian Toeplitz matrix

“…Circulant matrices with Fibonacci and Lucas numbers are discussed and their explicit determinants and inverses are proposed in [22]. The authors provided determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers in [2]. The explicit determinants of circulant and left circulant matrices including Tribonacci numbers and generalized Lucas numbers are shown based on Tribonacci numbers and generalized Lucas numbers in [17].…”