Handbook of Linear Algebra

The Handbook of Linear Algebra provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use handbook format. The esteemed international contributors guide you from the very elementary aspects of the subject to the frontiers of current research.

The book features an accessible layout of parts, chapters, and sections, with each section containing definition, fact, and example segments. The five main parts of the book encompass the fundamentals of linear algebra, combinatorial and numerical linear algebra, applications of linear algebra to various mathematical and nonmathematical disciplines, and software packages for linear algebra computations. Within each section, the facts (or theorems) are presented in a list format and include references for each fact to encourage further reading, while the examples illustrate both the definitions and the facts.

Linearization often enables difficult problems to be estimated by more manageable linear ones, making the Handbook of Linear Algebra essential reading for professionals who deal with an assortment of mathematical problems.

Table of Contents

Part I. Linear Algebra
Chapter 1. Vectors, Matrices, and Systems of Linear Equations
Chapter 2. Linear Independence, Span, and Bases
Chapter 3. Linear Transformations
Chapter 4. Determinants and Eigenvalues
Chapter 5. Inner Product Spaces, Orthogonal Projection, Least Squares, and Singular Value Decomposition
Chapter 6. Canonical Forms
Chapter 7. Unitary Similarity, Normal Matrices, and Spectral Theory
Chapter 8. Hermitian and Positive Definite Matrices
Chapter 9. Nonnegative Matrices and Stochastic Matrices
Chapter 10. Partitioned Matrices
Chapter 11. Functions of Matrices
Chapter 12. Quadratic, Bilinear, and Sesquilinear Forms
Chapter 13. Multilinear Algebra
Chapter 14. Matrix Equalities and Inequalities
Chapter 15. Matrix Perturbation Theory
Chapter 16. Pseudospectra
Chapter 17. Singular Values and Singular Value Inequalities
Chapter 18. Numerical Range
Chapter 19. Matrix Stability and Inertia
Chapter 20. Inverse Eigenvalue Problems
Chapter 21. Totally Positive and Totally Nonnegative Matrices
Chapter 22. Linear Preserver Problems
Chapter 23. Matrices over Integral Domains
Chapter 24. Similarity of Families of Matrices
Chapter 25. Max-Plus Algebra
Chapter 26. Matrices Leaving a Cone Invariant

Part II. Combinatorial Matrix Theory and Graphs
Chapter 27. Combinatorial Matrix Theory
Chapter 28. Matrices and Graphs
Chapter 29. Digraphs and Matrices
Chapter 30. Bipartite Graphs and Matrices
Chapter 31. Permanents
Chapter 32. D-Optimal Matrices
Chapter 33. Sign Pattern Matrices
Chapter 34. Multiplicity Lists for the Eigenvalues of Symmetric Matrices with a Given Graph
Chapter 35. Matrix Completion Problems
Chapter 36. Algebraic Connectivity

Part III. Numerical Methods
Chapter 37. Vector and Matrix Norms, Error Analysis, Efficiency, and Stability
Chapter 38. Matrix Factorizations and Direct Solution of Linear Systems
Chapter 39. Least Squares Solution of Linear Systems
Chapter 40. Sparse Matrix Methods
Chapter 41. Iterative Solution Methods for Linear Systems
Chapter 42. Symmetric Matrix Eigenvalue Techniques
Chapter 43. Unsymmetric Matrix Eigenvalue Techniques
Chapter 44. The Implicitly Restarted Arnoldi Methods
Chapter 45. Computation of the Singular Value Decomposition
Chapter 46. Computing Eigenvalues and Singular Values to High Relative Accuracy
Chapter 47. Fast Matrix Multiplication
Chapter 48. Structured Matrix Computations
Chapter 49. Large-Scale Matrix Computations

Part IV. Applications
Chapter 50. Linear Programming
Chapter 51. Semidefinite Programming
Chapter 52. Random Vectors and Linear Statistical Models
Chapter 53. Multivariate Statistical Analysis
Chapter 54. Markov Chains
Chapter 55. Differential Equations and Stability
Chapter 56. Dynamical Systems and Linear Algebra
Chapter 57. Control Theory
Chapter 58. Fourier Analysis
Chapter 59. Linear Algebra and Mathematical Physics
Chapter 60. Linear Algebra in Biomolecular Modeling
Chapter 61. Coding Theory
Chapter 62. Quantum Computation
Chapter 63. Information Retrieval and Web Search
Chapter 64. Signal Processing
Chapter 65. Geometry
Chapter 66. Some Applications of Matrices and Graphs in Euclidean Geometry
Chapter 67. Matrix Groups
Chapter 68. Group Representations
Chapter 69. Nonassociative Algebras
Chapter 70. Lie Algebras

Part V. Computational Software
Chapter 71. Matlab
Chapter 72. Linear Algebra in Maple
Chapter 73. Mathematica
Chapter 74. BLAS
Chapter 75. LAPACK
Chapter 76. Use of ARPACK and EIGS
Chapter 77. Summary of Software for Linear Algebra Freely Available on the Web