Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Symbols
- Part I Numerical Linear Algebra
- 1 Linear Operators and Matrices
- 2 The Singular Value Decomposition
- 3 Systems of Linear Equations
- 4 Norms and Matrix Conditioning
- 5 Linear Least Squares Problem
- 6 Linear Iterative Methods
- 7 Variational and Krylov Subspace Methods
- 8 Eigenvalue Problems
- Part II Constructive Approximation Theory
- Part III Nonlinear Equations and Optimization
- Part IV Initial Value Problems for Ordinary Differential Equations
- Part V Boundary and Initial Boundary Value Problems
- Appendix A Linear Algebra Review
- Appendix B Basic Analysis Review
- Appendix C Banach Fixed Point Theorem
- Appendix D A (Petting) Zoo of Function Spaces
- References
- Index
4 - Norms and Matrix Conditioning
from Part I - Numerical Linear Algebra
Published online by Cambridge University Press: 29 September 2022
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Symbols
- Part I Numerical Linear Algebra
- 1 Linear Operators and Matrices
- 2 The Singular Value Decomposition
- 3 Systems of Linear Equations
- 4 Norms and Matrix Conditioning
- 5 Linear Least Squares Problem
- 6 Linear Iterative Methods
- 7 Variational and Krylov Subspace Methods
- 8 Eigenvalue Problems
- Part II Constructive Approximation Theory
- Part III Nonlinear Equations and Optimization
- Part IV Initial Value Problems for Ordinary Differential Equations
- Part V Boundary and Initial Boundary Value Problems
- Appendix A Linear Algebra Review
- Appendix B Basic Analysis Review
- Appendix C Banach Fixed Point Theorem
- Appendix D A (Petting) Zoo of Function Spaces
- References
- Index
Summary
We introduce the spectral radius of a matrix, and study how it relates to induced matrix norms. We prove Gelfand’s theorem on the spectral radius. We introduce the condition number of a matrix, and use it to provide error estimates for the solution of a linear system under perturbations.
- Type
- Chapter
- Information
- Classical Numerical AnalysisA Comprehensive Course, pp. 73 - 87Publisher: Cambridge University PressPrint publication year: 2022