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
Abstract. Let H m×n be the space of m × n matrices over H, where H is the real quaternion algebra. Let A φ be the n × m matrix obtained by applying φ entrywise to the transposed matrix A T , where A ∈ H m×n and φ is a nonstandard involution of H. In this paper, some properties of the Moore-Penrose inverse of the quaternion matrix A φ are given. Two systems of mixed pairs of quaternion matrix Sylvester equationswhere Z is φ-Hermitian. Some practical necessary and sufficient conditions for the existence of a solution (X, Y, Z) to those systems in terms of the ranks and Moore-Penrose inverses of the given coefficient matrices are presented. Moreover, the general solutions to these systems are explicitly given when they are solvable. Some numerical examples are provided to illustrate the main results.
The generalized sequence of numbers is defined by Wn = pWn−1 + qWn−2 with initial conditions W0 = a and W1 = b for a, b, p, q ∈ Z and n ≥ 2, respectively. Let Wn = circ(W1, W2, . . . , Wn). The aim of this paper is to establish some useful formulas for the determinants and inverses of Wn using the nice properties of the number sequences. Matrix decompositions are derived for Wn in order to obtain the results. *
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