A Universal Identity for Powers in Quadratic Algebras and a Matrix Derivation of a Fibonacci Identity

TL;DR

The paper establishes a universal algebraic identity for powers in quadratic algebras, deriving a general matrix power formula and applying it to Fibonacci matrices to recover known identities through fundamental algebraic principles.

Contribution

It introduces a universal identity for powers in quadratic algebras and derives a matrix power formula that explains Fibonacci identities from basic algebraic concepts.

Findings

Derived a universal identity for powers in quadratic algebras

Obtained a general matrix power formula based on trace and determinant

Reproduced Fibonacci identities using algebraic principles

Abstract

We prove a universal identity for powers of elements in quadratic algebras, expressing x^m in terms of x and the identity. As a consequence, we obtain a general formula for powers of 2x2 matrices depending only on trace and determinant. Applying this to the Fibonacci matrix yields a binomial expansion formula for F_{nm}, recovering a recent identity of Vorobtsov. This shows that such identities arise from general algebraic principles rather than specific properties of Fibonacci numbers.