Abstract
Eigenvalues and the associated eigenvectors of an endomorphism of a vector space are defined and studied, as is the spectrum of an endomorphism. The characteristic polynomial of a matrix is considered and used to define the characteristic polynomial of the endomorphism of a finitely-generated vector space. This leads to the notions of geometric and algebraic multiplicities of eigenvalues. The minimal polynomial is then defined and studied. Among the theorems proven are the Cayley–Hamilton theorem and Burnside’s theorem. Von Mises’ algorithm for the computation of the dominant eigenvalue is introduced as an example of a iterative algorithm for eigenvalue computation.
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Notes
- 1.
The terms “eigenvalue” and “eigenvector” are due to Hilbert. Eigenvalues and eigenvectors are sometimes called characteristic values and characteristic vectors, respectively, based on terminology used by Cauchy. Sylvester coined the term “latent values” since, as he put it, such scalars are “latent in a somewhat similar sense as vapor may be said to be latent in water or smoke in a tobacco-leaf”.
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© 2012 Springer Science+Business Media B.V.
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Golan, J.S. (2012). Eigenvalues and Eigenvectors. In: The Linear Algebra a Beginning Graduate Student Ought to Know. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2636-9_12
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DOI: https://doi.org/10.1007/978-94-007-2636-9_12
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-2635-2
Online ISBN: 978-94-007-2636-9
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