On the matrix $pth$ root functions and generalized Fibonacci sequences

TL;DR

This paper explores polynomial representations of matrix pth root functions using generalized Fibonacci sequences, providing explicit formulas and numerical examples to enhance understanding of matrix root computations.

Contribution

It introduces a novel approach leveraging Fibonacci-Horner decomposition and properties of generalized Fibonacci sequences to derive explicit formulas for matrix pth roots.

Findings

Explicit formulas for matrix pth roots derived

Use of Fibonacci-Horner decomposition in matrix functions

Numerical examples illustrating the methods

Abstract

This study is devoted to the polynomial representation of the matrix $p$th root functions. The Fibonacci-H\"orner decomposition of the matrix powers and some techniques arisen from properties of generalized Fibonacci sequences, notably the Binet formula, serves as a triggering factor to provide explicit formulas for the matrix $p$th roots. Special cases and illustrative numerical examples are given.