Textbook in PDF format

After reading this book, students should be able to analyze computational problems in linear algebra such as linear systems, least squares- and eigenvalue problems, and to develop their own algorithms for solving them.
Since these problems can be large and difficult to handle, much can be gained by understanding and taking advantage of special structures. This in turn requires a good grasp of basic numerical linear algebra and matrix factorizations. Factoring a matrix into a product of simpler matrices is a crucial tool in numerical linear algebra, because it allows us to tackle complex problems by solving a sequence of easier ones.
The main characteristics of this book are as follows:
It is self-contained, only assuming that readers have completed first-year calculus and an introductory course on linear algebra, and that they have some experience with solving mathematical problems on a computer. The book provides detailed proofs of virtually all results. Further, its respective parts can be used independently, making it suitable for self-study.
The book consists of 15 chapters, divided into five thematically oriented parts. The chapters are designed for a one-week-per-chapter, one-semester course. To facilitate self-study, an introductory chapter includes a brief review of linear algebra.
Table of contents
A Short Review of Linear Algebra
Diagonally Dominant Tridiagonal Matrices; Three Examples
Gaussian Elimination and LU Factorizations
LDL* Factorization and Positive Definite Matrices
Orthonormal and Unitary Transformations
Eigenpairs and Similarity Transformations
The Singular Value Decomposition
Matrix Norms and Perturbation Theory for Linear Systems
Least Squares
The Kronecker Product
Fast Direct Solution of a Large Linear System
The Classical Iterative Methods
The Conjugate Gradient Method
Numerical Eigenvalue Problems
The QR Algorithm
Differentiation of Vector Functions