Jordan Canonical Form : Theory and Practice / by Steven H. Weintraub.

Format:

Book

Author/Creator:

Weintraub, Steven., Author.

Series:

Synthesis Lectures on Mathematics & Statistics, 1938-1751

Language:

English

Subjects (All):

Mathematics.

Statistics.

Engineering mathematics.

Engineering Mathematics.

Local Subjects:

Mathematics.

Statistics.

Engineering Mathematics.

Physical Description:

1 online resource (XI, 96 p.)

Edition:

1st ed. 2009.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2009.

Summary:

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 labeled eigenstructure picture (ℓESP) of A, determines P as well. We illustrate this algorithm with copious examples, and provide numerous exercises for the reader. Table of Contents: Fundamentals on Vector Spaces and Linear Transformations / The Structure of a Linear Transformation / An Algorithm for Jordan Canonical Form and Jordan Basis.

Contents:

Jordan Canonical Form

Solving Systems of Linear Differential Equations

Background Results: Bases, Coordinates, and Matrices

Properties of the Complex Exponential.

ISBN:

9783031023989

3031023986