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

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.