Matrix analysis and numerical linear algebra are two very active, and closely related, areas of research. Matrix analysis can be defined as the theory of matrices with a focus on aspects relevant to other areas of mathematics, while numerical linear algebra (also called matrix computations) is concerned with the construction and analysis of algorithms for solving matrix problems, as well as related topics such as problem sensitivity and rounding error analysis.
My article Numerical Linear Algebra and Matrix Analysis for The Princeton Companion to Applied Mathematics gives a selective overview of these two topics. The table of contents is as follows.
1 Nonsingularity and Conditioning 2 Matrix Factorizations 3 Distance to Singularity and Low-Rank Perturbations 4 Computational Cost 5 Eigenvalue Problems 5.1 Bounds and Localization 5.2 Eigenvalue Sensitivity 5.3 Companion Matrices and the Characteristic Polynomial 5.4 Eigenvalue Inequalities for Hermitian Matrices 5.5 Solving the Non-Hermitian Eigenproblem 5.6 Solving the Hermitian Eigenproblem 5.7 Computing the SVD 5.8 Generalized Eigenproblems 6 Sparse Linear Systems 7 Overdetermined and Underdetermined Systems 7.1 The Linear Least Squares Problem 7.2 Underdetermined Systems 7.3 Pseudoinverse 8 Numerical Considerations 9 Iterative Methods 10 Nonnormality and Pseudospectra 11 Structured Matrices 11.1 Nonnegative Matrices 11.2 M-Matrices 12 Matrix Inequalities 13 Library Software 14 Outlook
The article can be downloaded in pre-publication form as an EPrint.