Positive Definite Matrices
Positive Definite Matrices
复制标题
DOI：
10.1515/9781400827787
复制
发表时间：
2007
期刊：
影响因子：
--
通讯作者：
R. Bhatia
中科院分区：
文献类型：
--
作者：
R. Bhatia
这本书代表了第一次合成的相当大的机构的新的研究成正定矩阵。这些矩阵在非对易分析中的作用与正真实的数在经典分析中的作用相同。它们在广泛的学科中具有理论和计算用途，包括微积分，电气工程，统计学，物理学，数值分析，量子信息理论和几何学。通过详细的解释和权威和鼓舞人心的写作风格，Rajendra Bhatia仔细开发了在研究此类矩阵中具有广泛应用的一般技术。Bhatia介绍了泛函分析、算子理论、调和分析和微分几何中的几个关键主题--所有这些都围绕着正定矩阵的中心主题建立。他讨论了积极的和完全积极的线性映射，并提出了主要定理与简单和直接的证明。他检查矩阵手段及其应用，并展示了如何使用正定函数来推导算子不等式，他和其他人证明了近年来。他引导读者通过微分几何的流形正定矩阵，并解释了最近的工作几何平均数的几个矩阵。正定矩阵是一个信息丰富和有用的参考书数学家和其他研究人员和从业者。每章末尾的大量练习和注释也使其成为研究生课程的理想教科书。
This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.