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Jordan Canonical Form (JCF) is one of the most important, and useful, concepts in linear algebra. The JCF of a linear transformation, or of a matrix, encodes all of the structural information about that linear transformation, or matrix. This book is a careful development of JCF. After beginning with background material, we introduce Jordan Canonical Form and related notions: eigenvalues, (generalized) eigenvectors, and the characteristic and minimum polynomials.We decide the question of diagonalizability, and prove the Cayley-Hamilton theorem. Then we present a careful and complete proof of the fundamental theorem: Let V be a finite-dimensional vector space over the field of complex numbers C, and let T : V -. V be a linear transformation. Then T has a Jordan Canonical Form. This theorem has an equivalent statement in terms of matrices: Let A be a square matrix with complex entries. Then A is similar to a matrix J in Jordan Canonical Form, i.e., there is an invertible matrix P and a matrix J in Jordan Canonical Form with A = PJP-1.We further present an algorithm to find P and J , assuming that one can factor the characteristic polynomial of A. In developing this algorithm we introduce the eigenstructure picture (ESP) of a matrix, a pictorial representation that makes JCF clear. The ESP of A determines J , and a refinement, the labelled eigenstructure picture (ESP) of A, determines P as well.We illustrate this algorithm with copious examples, and provide numerous exercises for the reader.
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Previews available in: English
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Linear Algebras, Jordan algebras, EigenvaluesShowing 1 featured edition. View all 1 editions?
Edition | Availability |
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Jordan Canonical Form: theory and practice
2009, Morgan & Claypool Publishers
electronic resource :
in English
1608452514 9781608452514
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Edition Notes
Part of: Synthesis digital library of engineering and computer science.
Title from PDF t.p. (viewed on September 9, 2009).
Series from website.
Includes index.
Abstract freely available; full-text restricted to subscribers or individual document purchasers.
Also available in print.
Mode of access: World Wide Web.
System requirements: Adobe Acrobat reader.
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