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

Marko Pešović, Sonja Telebaković Onić

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.

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

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.

Paper Structure

This paper contains 7 sections, 18 theorems, 78 equations, 1 table.

Table of Contents

  1. Introduction
  2. The generalized $k$-Pell-Tribonacci sequence
  3. The $r$-circulant matrix
  4. Frobenius and entrywise $\ell_1$-norms
  5. Spectral norm
  6. Eigenvalues and determinant
  7. Acknowledgments

Key Result

Proposition 1

The $(n+1)$-th term of the sequence $\{P_{k,n}\}_{n\in\mathbb{N}_0}$ is given by the Binet-style formula: where $\alpha, \beta, \gamma$ are the distinct roots of the characteristic polynomial $\phi(x)$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (35)

  • Definition 1
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 25 more