The Numerical Jordan Form is the first book dedicated to exploring the algorithmic and computational methods for determining the Jordan form of a matrix, as well as addressing the numerical difficulties in finding it. Unlike the ‘pure’ Jordan form, the numerical Jordan form preserves its structure under small perturbations of the matrix elements so that its determination presents a well-posed computational problem. If this structure is well conditioned, it can be determined reliably in the presence of uncertainties and rounding errors.
This book addresses the form’s application in solving some important problems such as the estimation of eigenvalue sensitivity and computing the matrix exponential. Special attention is paid to the Jordan–Schur form of a matrix which, the author suggests, is not exploited sufficiently in the area of matrix computations. Since the mathematical objects under consideration can be sensitive to changes in the elements of the given matrix, the book also investigates the perturbation analysis of eigenvalues and invariant subspaces. This study is supplemented by a collection over 100 M-files suitable for MATLAB in order to implement the state-of-the art algorithms presented in the book for reducing a square matrix into the numerical Jordan form.
Researchers in the fields of numerical analysis and matrix computations and any scientists who utilise matrices in their work will find this book a useful resource, and it is also a suitable reference book for graduate and advance undergraduate courses in this subject area.
Contents:
- Preface
- List of Algorithms
- Notation
- Numerical Matrix Computations
- The Eigenvalue Problem
- The Eigenvalue Sensitivity Problem
- Numerical Solution of the Eigenvalue Problem
- Geometry of Jordan Forms
- Reduction into Jordan–Schur Form
- Reduction into Weyr and Jordan Forms
- Case Study 1: Eigenvalue Sensitivity Analysis
- Case Study 2: Computing the Matrix Exponential
- Appendix: Review of Linear Algebra and Matrix Analysis
- Bibliography
- Index
Readership: This book is suitable for practitioners and researchers in the fields of numerical analysis, matrix computations and applied mathematics, as well as scientists more broadly who use matrices in their work. It would also be suitable as a textbook or reference book for graduate and advanced undergraduate student courses related to numerical analysis, computation or numerical linear algebra, and matrix computations.
Key Features:
- The first book entirely devoted to the numerical computation of Jordan and Weyr canonical forms
- Considers for the first time in detail the Jordan–Schur form of a matrix and its application to the solution of several matrix problems
- Contains 100 numerical algorithms implemented as M-files for MATLAB as supplementary material
- Contains many numerical examples and figures illustrating the algorithms implementation and their behavior
- May be used as a reference on Numerical Matrix Computations or for a MSc course on Advanced Matrix Computations
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