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A direct proof of the existence of eigenvalues and eigenvectors by Weierstrass’s theorem. (English)
Am. Math. Mon. 120, No. 8, 741-746 (2013).
The existence of an eigenvector and an eigenvalues of a linear operator on a complex vector space is proved. The proof relies only on Weierstrass’ theorem (in the spirit of Argand’s proof), the definition of the inverse of a linear operator, and algebraic identities. Eigenvalues and eigenvectors are defined without reference to polynomials so the author presents a polynomial-free proof of the existence of eigenvectors and eigenvalues. It is not the shortest proof (see [{\it A. R. Schep}, Am. Math. Mon. 116, No. 1, 67‒68 (2009; Zbl 1175.30009)]).
Reviewer: Tin Yau Tam (Auburn)
Classification: H65