Jordan canonical form theory and practice
1. Fundamentals on vector spaces and linear transformations -- Bases and coordinates -- Linear transformations and matrices -- Some special matrices -- Polynomials in T and A -- Subspaces, complements, and invariant subspaces -- 2. The structure of a linear transformation -- Eigenvalues, eigenvector...
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| Format: | UnknownFormat |
| Sprache: | eng |
| Veröffentlicht: |
San Rafael, CA
Morgan & Claypool Publishers
2009
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| Schriftenreihe: | Synthesis lectures on mathematics and statistics
6 |
| Schlagworte: | |
| Online Zugang: | Inhaltstext Inhaltsverzeichnis |
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Tag hinzufügen | Zusammenfassung: | 1. Fundamentals on vector spaces and linear transformations -- Bases and coordinates -- Linear transformations and matrices -- Some special matrices -- Polynomials in T and A -- Subspaces, complements, and invariant subspaces -- 2. The structure of a linear transformation -- Eigenvalues, eigenvectors, and generalized eigenvectors -- The minimum polynomial -- Reduction to BDBUTCD form -- The diagonalizable case -- Reduction to Jordan Canonical Form -- Exercises -- 3. An algorithm for Jordan Canonical Form and Jordan Basis -- The ESP of a linear transformation -- The algorithm for Jordan Canonical Form -- The algorithm for a Jordan Basis -- Examples -- Exercises -- A. Answers to odd-numbered exercises -- Notation. 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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| Beschreibung: | Includes index |
| Beschreibung: | x, 96 Seiten Illustrationen |
| ISBN: | 9781608452507 978-1-60845-250-7 1608452506 1-60845-250-6 |