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Fibonacci vector and matrix p-norms

Francisco Salas-Molina

Abstract

This paper delves into vector and matrix norms of Fibonacci numbers. Two classes of Fibonacci vectors and a parametric p-norm are defined. From this definition, several properties of Fibonacci vector and matrix p-norms are described by varying parameter p. A closed-form expression is given to obtain the value of p, setting the difference between the p-norm and the infinite norm below a given threshold. A new class of symmetric k-Fibonacci matrix is defined such that a simple reorganization simplifies the computation of its p-norm. The analysis is extended to p-distances when considering the norm of the difference of two vectors (matrices) of the same size.

Fibonacci vector and matrix p-norms

Abstract

This paper delves into vector and matrix norms of Fibonacci numbers. Two classes of Fibonacci vectors and a parametric p-norm are defined. From this definition, several properties of Fibonacci vector and matrix p-norms are described by varying parameter p. A closed-form expression is given to obtain the value of p, setting the difference between the p-norm and the infinite norm below a given threshold. A new class of symmetric k-Fibonacci matrix is defined such that a simple reorganization simplifies the computation of its p-norm. The analysis is extended to p-distances when considering the norm of the difference of two vectors (matrices) of the same size.
Paper Structure (5 sections, 1 theorem, 45 equations)

This paper contains 5 sections, 1 theorem, 45 equations.

Table of Contents

  1. Introduction
  2. Fibonacci vector $p$-norms
  3. Fibonacci matrix $p$-norm
  4. Distances between vectors and matrices
  5. Conclusions

Key Result

Theorem 2.2

Given $\boldsymbol{q}_n$, there exists some $p < \infty$ such that $||\boldsymbol{q}_n||_p - ||\boldsymbol{q}_n||_{\infty} \leq \varepsilon$ for some value $\varepsilon>0$. The threshold value of $p$ is given by:

Theorems & Definitions (5)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Definition 4.1
  • Definition 4.2