Perspectives on the Spectral Theorem

  • Brian C. Hall
Part of the Graduate Texts in Mathematics book series (GTM, volume 267)


Suppose A is a self-adjoint n × n matrix, meaning that \(A_{kj} = \overline{A_{jk}}\) for all 1≤ j, kn. 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)


  1. [24].
    K. Hoffman, R. Kunze, Linear Algebra, 2nd edn. (Prentice-Hall, Englewood Cliffs, NJ, 1971)Google Scholar
  2. [34].
    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)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Brian C. Hall
    • 1
  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA

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