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

Abstract

A linear operator on a finite dimensional nonzero real vector space may not have an eigenvalue. However, corresponding to each such operator , there exist a pair of real numbers ( , ) and a nonzero vector such that [( − ) 2 + 2 ] ( ) = 0. 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 the Fundamental theorem of Algebra. 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.