Abstract
Selfadjoint endomorphisms of inner product spaces are defined and studied. Any selfadjoint endomorphism of a finitely-generated inner product space is shown to have a nonempty spectrum. The notion of orthogonal decomposition of an endomorphism is introduced. Selfadjoint endomorphisms of finitely-generated inner product spaces are shown to be orthogonally diagonalizable, with the converse true for spaces over the real numbers. Positive-definite endomorphisms are introduced and characterized. Application is made to Cholesky decompositions. Isometries of finitely-generated inner product spaces are studied and characterized.
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Systems of linear equations defined by positive-definite endomorphisms of ℝn first appear in Gauss’ work on least-squares approximation, which we will consider in a later chapter.
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© 2012 Springer Science+Business Media B.V.
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Golan, J.S. (2012). Selfadjoint Endomorphisms. In: The Linear Algebra a Beginning Graduate Student Ought to Know. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2636-9_17
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DOI: https://doi.org/10.1007/978-94-007-2636-9_17
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-2635-2
Online ISBN: 978-94-007-2636-9
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