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

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsMatrix Theory and Algorithms
