id: 06271811
dt: j
an: 2014d.00691
au: Van Schaftingen, Jean
ti: A direct proof of the existence of eigenvalues and eigenvectors by
Weierstrass’s theorem.
so: Am. Math. Mon. 120, No. 8, 741-746 (2013).
py: 2013
pu: Mathematical Association of America (MAA), Washington, DC
la: EN
cc: H65
ut: eigenvector; eigenvalue; linear operator; Weierstrass theorem;
polynomial-free proof
ci: Zbl 1175.30009
li: doi:10.4169/amer.math.monthly.120.08.741 arxiv:1109.6821
ab: 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)]).
rv: Tin Yau Tam (Auburn)