Abstract
Suppose A is a self-adjoint n × n matrix, meaning that \(A_{kj} = \overline{A_{jk}}\) for all 1≤ j, k ≤ n. Then a standard result in linear algebra asserts that there exist an orthonormal basis \(\{\mathbf{v}_{j}\}_{j=1}^{n}\) for \({\mathbb{C}}^{n}\) and real numbers λ 1,…,λ n such that \(A\mathbf{v}_{j} =\lambda _{j}\mathbf{v}_{j}\). (See Theorem 18 in Chap. 8 of [24] and Exercise 4 in Chap. 7)
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
K. Hoffman, R. Kunze, Linear Algebra, 2nd edn. (Prentice-Hall, Englewood Cliffs, NJ, 1971)
M. Reed, B. Simon, Methods of Modern Mathematical Physics. Volume I: Functional Analysis, 2nd edn. (Academic, San Diego, 1980). Volume II: Fourier Analysis, Self-Adjointness (Academic, New York, 1975). Volume III: Scattering Theory (Academic, New York, 1979). Volume IV: Analysis of Operators (Academic, New York, 1978)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Hall, B.C. (2013). Perspectives on the Spectral Theorem. In: Quantum Theory for Mathematicians. Graduate Texts in Mathematics, vol 267. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7116-5_6
Download citation
DOI: https://doi.org/10.1007/978-1-4614-7116-5_6
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-7115-8
Online ISBN: 978-1-4614-7116-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)