An introduction to applied matrix analysis Jin, Xiao-Qing

Table of Contents:

1.Introduction and Review
1.1.Basic symbols
1.2.Quadratic forms and positive definite matrices
1.2.1.Quadratic forms
1.2.2.Problems involving quadratic forms
1.2.3.Positive definite matrix
1.2.4.Other methods to determine the positive definiteness
1.3.Theorems for eigenvalues of symmetric matrices
1.4.Complex inner product spaces
1.5.Hermitian, unitary, and normal matrices
1.6.Kronecker product and Kronecker sum

2.Norms and Perturbation Analysis

2.1.Vector norms
2.2.Matrix norms
2.3.Perturbation analysis for linear systems
2.4.Error on floating point numbers

3.Least Squares Problems

3.1.Solution of LS problems
3.2.Perturbation analysis for LS problems
3.3.Orthogonal transformations
3.3.1.Householder reflections
3.3.2.Givens rotations
3.4.An algorithm based on QR factorization
3.4.1.QR factorization
3.4.2.A practical algorithm for LS problems

4.Generalized Inverses

4.1.Moore-Penrose generalized inverse
4.2.Basic properties
4.3.Relation to LS problems
4.4.Other generalized inverses

5.Conjugate Gradient Method

5.1.Steepest descent method
5.1.1.Steepest descent method
5.1.2.Convergence rate
5.2.Conjugate gradient method
5.2.1.Conjugate gradient method
5.2.2.Basic properties
5.2.3.Practical CG method
5.3.Preconditioning technique

6.Optimal and Superoptimal Preconditioners

6.1.Introduction to optimal preconditioner
6.1.1.Circulant matrix
6.1.2.Optimal preconditioner
6.2.Linear operator cU
6.2.1.Algebraic properties
6.2.2.Geometric properties
6.3.Stability
6.4.Superoptimal preconditioner
6.5.Spectral relation of preconditioned matrices

7.Optimal Preconditioners for Functions of Matrices

7.1.Optimal preconditioners for matrix exponential
7.2.Optimal preconditioners for matrix cosine and matrix sine
7.3.Optimal preconditioners for matrix logarithm

8.Bottcher-Wenzel Conjecture and Related Problems

8.1.Introduction to Bottcher-Wenzel conjecture
8.2.The proof of Bottcher-Wenzel conjecture
8.3.Maximal pairs of the inequality
8.4.Other related problems
8.4.1.The use of other norms in the inequality
8.4.2.The sharpening of the inequality
8.4.3.The extension to other products similar to the commutator.

It is well known that most problems in science and engineering eventually progress into matrix problems. This book gives an elementary introduction to applied matrix theory and it also includes some new results obtained in recent years.

The book consists of eight chapters. It includes perturbation and error analysis; the conjugate gradient method for solving linear systems; preconditioning techniques; and least squares algorithms based on orthogonal transformations, etc. The last two chapters include some latest development in the area. In Chap. 7, we construct optimal preconditioners for functions of matrices. More precisely, let f be a function of matrices. Given a matrix A, there are two choices of constructing optimal preconditioners for f(A). Properties of these preconditioners are studied for different functions. In Chap. 8, we study the Bottcher–Wenzel conjecture and discuss related problems.

This is a textbook for senior undergraduate or junior graduate students majoring in science and engineering. The material is accessible to students who, in various disciplines, have basic linear algebra, calculus, numerical analysis, and computing knowledge. The book is also useful to researchers in computational science who are interested in applied matrix theory.

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