Simple proof of existence of a complex eigenvalue of a complex square matrix …and yet another proof of the fundamental theorem of algebra with linear algebra

Abstract

We give a simple proof of the existence of a complex eigenvalue of real or complex square matrices with Linear Algebra accessible to students beginning a first course on the subject, namely using just the Cauchy–Schwarz inequality in , linear independence and the matrix product, together with the Weierstrass extreme value theorem. It yields an equally simple proof of the Fundamental Theorem of Algebra with Linear Algebra.