Ercan Altınışık

GCD matrices, posets, and nonintersecting paths

We show that with any finite partially ordered set P (which need not be a lattice) one can associate a matrix whose determinant factors nicely. This was also noted by D.A. Smith, although his proof uses manipulations in the incidence algebra of P while ours is combinatorial, using nonintersecting paths in a digraph. As corollaries, we obtain new proofs for and generalizations of a number of results in the literature about GCD matrices and their relatives.

On the representation of k-generalized Fibonacci and Lucas numbers

Abstract In this paper we give some determinantal and permanental representations of k-generalized Fibonacci and Lucas numbers. We obtain the Binet’s formula for these sequences by using our representations.

Determinant and Inverse of Meet and Join Matrices

We define meet and join matrices on two subsets X and Y of a lattice (P, ) with respect to a complex-valued function f on P by (X ,Y) f = ( f (xi ∧ yi)) and [X ,Y] f = ( f (xi ∨ yi)), respectively. We present expressions for the determinant and the inverse of (X ,Y) f and [X ,Y] f , and as special cases we obtain several new and known formulas for the determinant and the inverse of the usual meet and join matrices (S) f and [S] f .

A proof of a conjecture on monotonic behavior of the largest eigenvalue of a number-theoretic matrix

In this study we investigate the monotonic behavior of the largest eigenvalue of the n×n matrix EnTEn, where the i j− entry of En is 1 if j|i and 0 otherwise and hence we present a proof of a part of the Mattila-Haukkanen conjecture [16]. MSC2010. 15A18, 15A42, 11A25

Determinants and inverses of circulant matrices with complex Fibonacci numbers

Abstract Let ℱn = circ (︀F*1 , F*2, . . . , F*n︀ be the n×n circulant matrix associated with complex Fibonacci numbers F*1, F*2, . . . , F*n. In the present paper we calculate the determinant of ℱn in terms of complex Fibonacci numbers. Furthermore, we show that ℱn is invertible and obtain the entries of the inverse of ℱn in terms of complex Fibonacci numbers.