True-pairs of Real Linear Operators and Factorization of Real Polynomials

TL;DR

The paper introduces the concept of true-pairs of real linear operators and proves their existence without relying on fundamental algebraic theorems, leading to a polynomial factorization result that avoids complex numbers.

Contribution

It presents a novel inductive proof of the existence of true-pairs for real linear operators, enabling polynomial factorization without complex analysis.

Findings

Every real linear operator has a true-pair.

Real polynomials can be factored into linear and quadratic factors without complex numbers.

Provides an inductive proof avoiding the Fundamental Theorem of Algebra and Cayley-Hamilton theorem.

Abstract

A linear operator on a finite dimensional nonzero real vector space may not have an eigenvalue. We define a related notion of a true-pair of a linear operator, and then show that each linear operator on a finite dimensional nonzero real vector space has a true-pair. This is usually proved by using the Fundamental theorem of algebra and Cayley-Hamilton theorem. We construct an inductive proof of this fact without using these theorems. From this we deduce that a polynomial with real coefficients can be written as a product of linear factors and quadratic factors with negative discriminant. It thus gives a proof of the latter fact about polynomials with real coefficients, which does not use complex numbers.