Numerical Linear AlgebraThis 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. 
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This is an excellent book on numerical linear algebra, a very good textbook for a senior undergraduate course. I like the writing style and have been enjoying the reading. Very often it explains "why", not just gives "what" and "how". Highly recommended.
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The strength of this book is in the conceptual discussions. This isn't the book to use to learn the mechanics of the methods described. It's one of my three favorite numerical linear algebra books.
Contents
Fundamentals  8 
MatrixVector Multiplication  9 
Orthogonal Vectors and Matrices  11 
Norms  17 
The Singular Value Decomposition  25 
More on the SVD  32 
QR Factorization and Least Squares  39 
Projectors  41 
Cholesky Factorization  172 
Eigenvalues  179 
Eigenvalue Problems  181 
Overview of Eigenvalue Algorithms  190 
Reduction to Hessenberg or Tridiagonal Form  196 
Rayleigh Quotient Inverse Iteration  202 
QR Algorithm without Shifts  211 
QR Algorithm with Shifts  219 
QR Factorization  48 
Gram Schmidt Orthogonalization  56 
MATLAB  63 
Householder Triangularization  69 
Least Squares Problems  77 
Conditioning and Stability  87 
Conditioning and Condition Numbers  89 
Floating Point Arithmetic  97 
Stability  102 
More on Stability  108 
Stability of Householder Triangularization  114 
Stability of Back Substitution  121 
Conditioning of Least Squares Problems  129 
Stability of Least Squares Algorithms  137 
Systems of Equations  145 
Gaussian Elimination  147 
Pivoting  155 
Stability of Gaussian Elimination  163 
Other Eigenvalue Algorithms  225 
Computing the SVD  234 
Iterative Methods  241 
Overview of Iterative Methods  243 
The Arnoldi Iteration  250 
How Arnoldi Locates Eigenvalues  257 
GMRES  266 
The Lanczos Iteration  276 
From Lanczos to Gauss Quadrature  285 
Conjugate Gradients  293 
Biorthogonalization Methods  303 
Preconditioning  313 
Appendix The Definition of Numerical Analysis  321 
Notes  329 
343  
353  
Common terms and phrases
accuracy applied approximation Arnoldi iteration backward stable bidiagonalization CG iteration Cholesky factorization Cmxm Cmxn coefficients column spaces columns of Q condition number convergence corresponding defined denote diagonal matrix dimension eigenvalue decomposition eigenvalue problem eigenvectors emachine entries example Exercise Figure floating point arithmetic flops follows formula full rank function Gaussian elimination geometric GMRES GramSchmidt Hessenberg Householder triangularization idea illconditioned introduce zeros inverse iteration Krylov subspace Lanczos iteration least squares problem m x m m x n matrix machine mathematical MATLAB matrix norm minimal multiplication nonsingular nonzero normal equations numerical analysis numerical linear algebra O(emachine operation count orthogonal projector orthonormal partial pivoting perturbations preconditioner QR algorithm random matrices Rayleigh quotient Rayleigh quotient iteration reduced QR factorization result Ritz values rounding errors satisfying sequence singular values singular vectors solution solve stability step Suppose symmetric system of equations Theorem tridiagonal unitary matrix unstable uppertriangular