Abstract
Unitary endomorphisms of an inner product space are introduced and studied. In particular, unitary and orthogonal matrices are considered. Many examples are considered, among them Householder matrices and special orthogonal matrices. Unitarily-similar matrices are defined and Schur’s Theorem is proven. Normal endomorphisms of an inner product space are also considered and their properties proven, leading up to a proof of the Spectral Decomposition Theorem and the Singular Value Decomposition Theorem.
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Notes
- 1.
Polar decompositions were first studied by the French engineer Léon Autonne at the beginning of the twentieth century.
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© 2012 Springer Science+Business Media B.V.
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Golan, J.S. (2012). Unitary and Normal Endomorphisms. In: The Linear Algebra a Beginning Graduate Student Ought to Know. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2636-9_18
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DOI: https://doi.org/10.1007/978-94-007-2636-9_18
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-2635-2
Online ISBN: 978-94-007-2636-9
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