This is a concise, insightful introduction to the field of numerical linear algebra. The clarity and eloquence of the presentation make it popular with teachers and students alike. The text aims to expand the reader's view of the field and to present standard material in a novel way. All of the most important topics in the field are covered with a fresh perspective, including iterative methods for systems of equations and eigenvalue problems and the underlying principles of conditioning and stability. Presentation is in the form of 40 lectures, which each focus on one or two central ideas. The unity between topics is emphasized throughout, with no risk of getting lost in details and technicalities. The book breaks with tradition by beginning with the QR factorization - an important and fresh idea for students, and the thread that connects most of the algorithms of numerical linear algebra.

I Fundamentals

Lecture 1 Matrix-Vector Multiplication

Lecture 2 Orthogonal Vectors and Matrices

Lecture 3 Norms

Lecture 4 The Singular Value Decomposition

Lecture 5 More on the SVD

II QR Factorization and Least Squares

Lecture 6 Projectors

Lecture 7 QR Factorization . . . . . . . . .

Lecture 8 Gram-Schmidt Orthogonalization

Lecture 9 MATLAB . . . . . . . . . . .

Lecture 10 Householder Triangularization

Lecture 11 Least Squares Problems

III Conditioning and Stability

Lecture 12 Conditioning and Condition Numbers

Lecture 13 Floating Point Arithmetic

Lecture 14 Stability

Lecture 15 More on Stability

Lecture 16 Stability of Householder Triangularization

Lecture 17 Stability of Back Substitution

Lecture 18 Conditioning of Least Squares Problems

Lecture 19 Stability of Least Squares Algorithms

IV Systems of Equations

Lecture 20 Gaussian Elimination

Lecture 21 Pivoting

Lecture 22 Stability of Gaussian Elimination

Lecture 23 Cholesky Factorization

V Eigenvalues

Lecture 24 Eigenvalue Problems

Lecture 25 Overview of Eigenvalue Algorithms

Lecture 26 Reduction to Hessenberg or Tridiagonal Form

Lecture 27 Rayleigh Quotient, Inverse Iteration

Lecture 28 QR Algorithm without Shifts

Lecture 29 QR Algorithm with Shifts

Lecture 30 Other Eigenvalue Algorithms

Lecture 31 Computing the SVD

VI Iterative Methods

Lecture 32 Overview of Iterative Methods

Lecture 33 The Arnoldi Iteration

Lecture 34 How Arnoldi Locates Eigenvalues

Lecture 35 GMRES

Lecture 36 The Lanczos Iteration

Lecture 37 From Lanczos to Gauss Quadrature

Lecture 38 Conjugate Gradients

Lecture 39 Biorthogonalization Methods

Lecture 40 Preconditioning

Review

' The authors are to be congratulated on producing a fresh and lively introduction to a fundamental area of numerical analysis.' G. W. Stewart, Mathematics of Computation

'…Each lecture in the textbook is pleasantly written in a conversational style and concludes with a set of related exercises. This low-cost textbook emphasizes many important and relevant topics in numerical linear algebra and seems ideal for a graduate course as long as it is accompanied by a textbook with more mathematical details.' Ricardo D. Fierro, SIAM Review

'Trefethen and Bau clear the dark clouds from numerical problems associated with factoring matrices, solving linear equations, and finding eigenvalues.' P. Cull, CHOICE

'Just exactly what I might have expected - an absorbing look at the familiar topics through the eyes of a master expositor. I have been reading it and learning a lot.' Paul Saylor, University of Illinois, Urbana-Champaign

'A beautifully written textbook offering a distinctive and original treatment. It will be of use to all who teach or study the subject.' Nicholas J. Higham, Professor of Applied Mathematics, University of Manchester

"This is a beautifully written book which carefully brings to the reader the important issues connected with the computational issues in matrix computations. The authors show a broad knowledge of this vital area and make wonderful connections to a variety of problems of current interest. The book is like a delicate soufflé --- tasteful and very light." -Gene Golub, Stanford University.

"I have used Numerical Linear Algebra in my introductory graduate course and I have found it to be almost the perfect text to introduce mathematics graduate students to the subject. I like the choice of topics and the format: a sequence of lectures. Each chapter (or lecture) carefully builds upon the material presented in previous chapters, providing new concepts in a very clear manner. Exercises at the end of each chapter reinforce the concepts, and in some cases introduce new ones. …The emphasis is on the mathematics behind the algorithms, in the understanding of why the algorithms work. …The text is sprinkled with examples and explanations, which keep the student focused." -Daniel B. Szyld, Department of Mathematics, Temple University.

"…This is an ideal book for a graduate course in numerical linear algebra (either in mathematics or in computer science departments); it presents the topics in such a way that background material comes along with the course. …I will use it again next time I teach this course!" -Suely Oliveira, Texas A&M University.