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Linear Algebra pp 246-273 | Cite as

The spectral theorem and quadratic forms

  • Larry Smith
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

So far in our study of linear transformations we have concentrated on trying to find conditions that assure the matrix of the transformation of a particular form. We have not asked the related question of studying properties of those transformations whose matrix is assumed to have a particularly simple form. There is in fact a good reason for this and it is tied up with our work of the last chapter. For example we might propose to study those linear transformations whose matrix is symmetric. We would therefore like to introduce the following:
  • Proposed Definition. Let T: VV be a linear transformation. We say that T is symmetric iff there is a basis {A1,…A n } for V such that the matrix of T relative to this basis is symmetric.

Keywords

Quadratic Form Orthonormal Basis Linear Transformation Product Space Characteristic Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag, New York Inc. 1978

Authors and Affiliations

  • Larry Smith
    • 1
    • 2
  1. 1.Mathematisches InstitutUniversität GöttingenGöttingenWest Germany
  2. 2.Department of MathematicsIndiana UniversityBloomingtonUSA

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