巴蒂亚编著的《矩阵分析》旨在为读者提供泛函分析的精髓矩阵分析。算子理论，算子代数，数学物理和数值分析专业的研究生和科研人员将会对本书感兴趣。本书可以作为高等线性代数和矩阵分析方向的研究生基础教程，也可以作为算子理论和数值分析方向的补充教程，包括的核心思想有最优化理论，特征值的变分原理，算子单调性和凸分析，矩阵函数的扰动和矩阵不等式。这些内容大多数都是第一次以《矩阵分析》中这么独特的方式讲述。读者将会从书中学到很多强大的工具、广泛的应用技巧以及和数学专业其他领域之间的联系。矩阵不等式使得《矩阵分析》对数值分析，数学物理和算子理论专业中学生，科研人员的参考价值凸显。

preface

i a review of linear algebra

 i.1 vector spaces and inner product spaces

 i.2 linear operators and matrices

 i.3 direct sums

 i.4 tensor products

 i.5 symmetry classes

 i.6 problems

 i.7 notes and references

ii majorisation and doubly stochastic matrices

 ii.1 basic notions

 ii.2 birkhoff's theorem

 ii.3 convex and monotone functions

 ii.4 binary algebraic operations and majorisation

 ii.5 problems

 ii.6 notes and references

iii variational principles for eigenvalues

 ili.1 the minimax principle for eigenvalues

 iii.2 weyl's inequalities

 iii.3 wielandt's minimax principle

 iii.4 lidskii's theorems

 iii.5 eigenvalues of real parts and singular values

 iii.6 problems

 iii.7 notes and references

iv symmetric norms

 iv.1 norms on cn

 iv.2 unitarily invariant norms on operators on cn

 iv.3 lidskii's theorem (third proof)

 iv.4 weakly unitarily invariant norms

 iv.5 problems

 iv.6 notes and references

v operator monotone and operator convex functions

 v.1 definitions and simple examples

 v.2 some characterisations

 v.3 smoothness properties

 v.4 loewner's theorems

 v.5 problems

 v.6 notes and references

vi spectral variation of normal matrices

 vi.1 continuity of roots of polynomials

 vi.2 hermitian and skew-hermitian matrices

 vi.3 estimates in the operator norm

 vi.4 estimates in the probenius norm

 vi.5 geometry and spectral variation: the operator norm

 vi.6 geometry and spectral variation: wui norms

 vi.7 some inequalities for the determinant

 vi.8 problems

 vi.9 notes and references

vii perturbation of spectral subspaces of normal matrices

 vii.1 pairs of subspaces

 vii.2 the equation ax - xb = y

 vii.3 perturbation of eigenspaces

 vii.4 a perturbation bound for eigenvalues

 vii.5 perturbation of the polar factors

 vii.6 appendix: evaluating the (fourier) constants

 vii.7 problems

 vii.8 notes and references

viii spectral variation of nonnormal matrices

 viii.1 general spectral variation bounds

 viii.4 matrices with real eigenvalues

 viii.5 eigenvalues with symmetries

 viii.6 problems

 viii.7 notes and references

ix a selection of matrix inequalities

 ix.1 some basic lemmas

 ix.2 products of positive matrices

 ix.3 inequalities for the exponential function

 ix.4 arithmetic-geometric mean inequalities

 ix.5 schwarz inequalities

 ix.6 the lieb concavity theorem

 ix.7 operator approximation

 ix.8 problems

 ix.9 notes and references

x perturbation of matrix functions

 x.1 operator monotone functions

 x.2 the absolute value

 x.3 local perturbation bounds

 x.4 appendix: differential calculus

 x.5 problems

 x.6 notes and references

references

index