Jordan Canonical Form theory and practice /

• Main Author: Weintraub, Steven H.
• Format: Electronic
• Language: English
• Published: San Rafael, Calif. (1537 Fourth Street, San Rafael, CA 94901 USA) : Morgan & Claypool Publishers, c2009.
• Series: Synthesis lectures on mathematics and statistics (Online), # 6.
• Subjects: Jordan algebras.; Algebras, Linear.; Eigenvalues.
• Online Access: Abstract with links to full text
• Item Description: 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.
• Physical Description: 1 electronic text (x, 96 p. : ill.) : digital file.; Also available in print.
• Format: Mode of access: World Wide Web.; System requirements: Adobe Acrobat reader.
• ISBN: 9781608452514 (electronic bk.)
• ISSN: 1938-1751
• Access: Abstract freely available; full-text restricted to subscribers or individual document purchasers.

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.