• An introduction to applied matrix analysis
• 點閱：12
• 並列題名：應用矩陣分析導論
• 作者： 金小慶, 黃鍚榮[著]
• 出版社：高等教育出版社
• 出版年：2016[民105]
• 集叢名：Series in contemporary applied mathematics:20
• ISBN：9787040449945
• 格式：PDF
• 附註：內容為英文

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.

• Preface(第vii頁)
• 1. Introduction and Review(第1頁)
• 1.1 Basic symbols(第1頁)
• 1.2 Quadratic forms and positive definite matrices(第2頁)
• 1.3 Theorems for eigenvalues of symmetric matrices(第6頁)
• 1.4 Complex inner product spaces(第10頁)
• 1.5 Hermitian , unitary , and normal matrices(第11頁)
• 1.6 Kronecker product and Kronecker sum(第13頁)
• 2. Norms and Perturbation Analysis(第17頁)
• 2.1 Vector norms(第17頁)
• 2.2 Matrix norms(第18頁)
• 2.3 Perturbation analysis for linear systems(第27頁)
• 2.4 Error on floating point numbers(第32頁)
• 3. Least Squares Problems(第37頁)
• 3.1 Solution of LS problems(第37頁)
• 3.2 Perturbation analysis for LS problems(第41頁)
• 3.3 Orthogonal transformations(第42頁)
• 3.4 An algorithm based on QR factorization(第45頁)
• 4. Generalized Inverses(第51頁)
• 4.1 Moore - Penrose generalized inverse(第51頁)
• 4.2 Basic properties(第53頁)
• 4.3 Relation to LS problems(第54頁)
• 4.4 Other generalized inverses(第56頁)
• 5.1 Steepest descent method(第61頁)
• 5.3 Preconditioning technique(第72頁)
• 6. Optimal and Superoptimal Preconditioners(第75頁)
• 6.1 Introduction to optimal preconditioner(第75頁)
• 6.2 Linear operator cU(第80頁)
• 6.3 Stability(第85頁)
• 6.4 Superoptimal preconditioner(第87頁)
• 6.5 Spectral relation of preconditioned matrices(第94頁)
• 7. Optimal Preconditioners for Functions of Matrices(第99頁)
• 7.1 Optimal preconditioners for matrix exponential(第99頁)
• 7.2 Optimal preconditioners for matrix cosine and matrix sine(第104頁)
• 7.3 Optimal preconditioners for matrix logarithm(第107頁)
• 8. Böttcher - Wenzel Conjecture and Related Problems(第111頁)
• 8.1 Introduction to Böttcher - Wenzel conjecture(第111頁)
• 8.2 The proof of Böttcher - Wenzel conjecture(第112頁)
• 8.3 Maximal pairs of the inequality(第116頁)
• 8.4 Other related problems(第121頁)
• Bibliography(第125頁)
• Index(第129頁)