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Inner-Product Spaces

  • Anthony W. Knapp
Part of the Cornerstones book series (COR)

Abstract

This chapter investigates the effects of adding the additional structure of an inner product to a finite-dimensional real or complex vector space.

Section 1 concerns the effect on the vector space itself, defining inner products and their corresponding norms and giving a number of examples and formulas for the computation of norms. Vector-space bases that are orthonormal play a special role.

Section 2 concerns the effect on linear maps. The inner product makes itself felt partly through the notion of the adjoint of a linear map. The section pays special attention to linear maps that are self-adjoint, i.e., are equal to their own adjoints, and to those that are unitary, i.e., preserve norms of vectors.

Section 3 proves the Spectral Theorem for self-adjoint linear maps on finite-dimensional innerproduct spaces. The theorem says in part that any self-adjoint linear map has an orthonormal basis of eigenvectors. The Spectral Theorem has several important consequences, one of which is the existence of a unique positive semidefinite square root for any positive semidefinite linear map. The section concludes with the polar decomposition, showing that any linear map factors as the product of a unitary linear map and a positive semidefinite one.

Keywords

Orthonormal Basis Hermitian Matrix Vector Subspace Polar Decomposition Complex Vector Space 
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

© Anthony W. Knapp 2006

Authors and Affiliations

  1. 1.East SetauketUSA

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