Abstract
Chapter 4 begins with the definition of eigenvalues, eigenvectors, eigenspaces, annihilator, and characteristic polynomials, etc. Then linear transformations of a complex or real vector space to itself are investigated in greater detail. In this chapter we consider the case in which a linear transformation is diagonalizable. Namely, for a complex vector space, we obtain necessary and sufficient conditions for a linear transformation to be diagonalizable. Analogously, for a real vector space, we obtain necessary and sufficient conditions for a linear transformation to be block-diagonalizable. Finally, the notion of orientation of a real vector space is considered.
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Notes
- 1.
For the general notion of the discriminant of a polynomial, see, for instance, Polynomials, by Victor V. Prasolov, Springer 2004.
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© 2012 Springer-Verlag Berlin Heidelberg
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Shafarevich, I.R., Remizov, A.O. (2012). Linear Transformations of a Vector Space to Itself. In: Linear Algebra and Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30994-6_4
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DOI: https://doi.org/10.1007/978-3-642-30994-6_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-30993-9
Online ISBN: 978-3-642-30994-6
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