On the Algebraic Properties of r-circulant Matrices Associated with Generalized k-Pell-Tribonacci Numbers

TL;DR

This paper investigates the algebraic properties of r-circulant matrices constructed from generalized k-Pell-Tribonacci numbers, deriving explicit formulas for norms, eigenvalues, and determinants, and establishing bounds for spectral norms.

Contribution

It provides new explicit formulas and bounds for r-circulant matrices associated with generalized k-Pell-Tribonacci sequences, extending and sharpening previous results.

Findings

Explicit formulas for Frobenius and -norms

Closed-form eigenvalues and determinants

Sharper bounds for spectral norms

Abstract

This study examines the properties of an r-circulant matrix whose entries are defined by the generalized k-Pell-Tribonacci sequence {P_k,n}. Explicit expressions are derived for the Frobenius (Euclidean) norm and the entrywise \ell_1-norm, together with closed-form formulas for the eigenvalues and the determinant of the matrix. Furthermore, upper and lower bounds for the spectral norm are established, yielding results that generalize previously reported ones corresponding to particular sequences while also providing sharper bounds for the considered norms.