Linear Algebra pp 251-278 | Cite as

Complex Vector Spaces

  • Jin Ho Kwak
  • Sungpyo Hong

Abstract

A real matrix has real coefficients in its characteristic polynomial, but the eigenvalues may fail to be real. For instance, the matrix \(A = \left[ {\begin{array}{*{20}{l}} 1&{ - 1} \\ 1&1 \end{array}} \right]\) has no real eigenvalues, but it has the complex eigenvalues λ= 1 ± i. Thus, it is indispensable to work with complex numbers to find the full set of eigenvalues and eigenvectors. Therefore, it is natural to extend the concept of real vector spaces to that of complex vector spaces, and then develop the basic properties of complex vector spaces. With this extension, all the square matrices of order n will have n eigenvalues.

Keywords

Orthonormal Basis Spectral Decomposition Triangular Matrix Real Eigenvalue Hermitian Matrix 
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 Science+Business Media New York 1997

Authors and Affiliations

  • Jin Ho Kwak
    • 1
  • Sungpyo Hong
    • 1
  1. 1.Department of MathematicsPohang University of Science and TechnologyPohangThe Republic of Korea

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