Jordan Canonical Form : Theory and Practice PDF
by Steven H. Weintraub
Part of the Synthesis Lectures on Mathematics & Statistics series
Description
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
Information
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Download - Immediately Available
- Format:PDF
- Publisher:Springer International Publishing
- Publication Date:01/06/2022
- Category:
- ISBN:9783031023989
Other Formats
- Paperback / softback from £27.99
Information
-
Download - Immediately Available
- Format:PDF
- Publisher:Springer International Publishing
- Publication Date:01/06/2022
- Category:
- ISBN:9783031023989
