Determinants and inverses ofr-circulant matrices associated with a number sequence

Abstract

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. *

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

“…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].…”

“…with initial conditions W 0 = a and W 1 = b, here a, b, p, q ∈ Z [3,4]. Let α and β be the roots of x 2 − px + q = 0.…”

Some norm inequalities for special Gram matrices

“…For example, Türkmen and Civciv have established some norm inequalities on circulant matrices with lucas numbers [1]. S.Solak has studied the norms of circulant matrices with bonacci and lucas numbers [2], Bozkurt and Tam have obtained some results belong to determinants and inverses of r-circulant matrices associated with a number sequence [3] and S.Shen and J.Cen have made a similar study by using the same special matrix with k-bonacci and k-lucas numbers [4]. For more properties, formulas belong to the Fibonacci, Lucas and Pell numbers (see, e.g., [11,12]).…”

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. Namely, they calculated the determinant of W = [7].…”