Positive Definite Matrices

Rajendra Bhatia; Rajendra Bhatia

Buch

Positive Definite Matrices

Rajendra Bhatia und Rajendra Bhatia

Sprache: Englisch

Veröffentlicht/Copyright: 2009

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Über dieses Buch

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.

Information zu Autoren / Herausgebern

Rajendra Bhatia is Professor of Mathematics at the Indian Statistical Institute in New Delhi. He is the author of five books, including Matrix Analysis.

Rezensionen

"This is an outstanding book. Its exposition is both concise and leisurely at the same time."---Jaspal Singh Aujla, Zentralblatt MATH --- "There is no obvious competitor for Bhatia's book, due in part to its focus, but also because it contains some very recent material drawn from research articles. Beautifully written and intelligently organised, Positive Definite Matrices is a welcome addition to the literature. Readers who admired his Matrix Analysis will no doubt appreciate this latest book of Rajendra Bhatia."---Douglas Farenick, Image --- "Written by an expert in the area, the book presents in an accessible manner a lot of important results from the realm of positive matrices and of their applications. . . . The book can be used for graduate courses in linear algebra, or as supplementary material for courses in operator theory, and as a reference book by engineers and researchers working in the applied field of quantum information."---S. Cobzas, Studia Universitatis Babes-Bolyai, Mathematica

Fachgebiete

Frontmatter Öffentlich zugänglich PDF downloaden	i
Contents Öffentlich zugänglich PDF downloaden	v
Preface Erfordert eine Authentifizierung Nicht lizenziert Lizenziert	vii
Chapter One. Positive Matrices Erfordert eine Authentifizierung Nicht lizenziert Lizenziert	1
Chapter Two. Positive Linear Maps Erfordert eine Authentifizierung Nicht lizenziert Lizenziert	35
Chapter Three. Completely Positive Maps Erfordert eine Authentifizierung Nicht lizenziert Lizenziert	65
Chapter Four. Matrix Means Erfordert eine Authentifizierung Nicht lizenziert Lizenziert	101
Chapter Five. Positive Definite Functions Erfordert eine Authentifizierung Nicht lizenziert Lizenziert	141
Chapter Six. Geometry of Positive Matrices Erfordert eine Authentifizierung Nicht lizenziert Lizenziert	201
Bibliography Erfordert eine Authentifizierung Nicht lizenziert Lizenziert	237
Index Erfordert eine Authentifizierung Nicht lizenziert Lizenziert	247
Notation Erfordert eine Authentifizierung Nicht lizenziert Lizenziert	253

Informationen zur Veröffentlichung

Seiten und Bilder/Illustrationen im Buch

eBook veröffentlicht am:

10. Januar 2009

eBook ISBN:

9781400827787

Auflage:

Course Book

Seiten und Bilder/Illustrationen im Buch

https://doi.org/10.1515/9781400827787

eBook ISBN: 9781400827787

Schlagwörter für dieses Buch

Theorem; Linear map; Positive-definite matrix; Hermitian matrix; Linear algebra; Geometric mean; Hilbert space; Hadamard product (matrices); Block matrix; Eigenvalues and eigenvectors; Monotonic function; Special case; Existential quantification; Convex set; Positive map; Completely positive map; Mathematics; Matrix analysis; Self-adjoint operator; Unitary matrix; Sign (mathematics); Logarithmic mean; Orthonormal basis; Square root; Summation; Complex number; Positive element; Scientific notation; Positive-definite function; Schur complement; Lecture; Density matrix; Spectral theorem; Addition; Geometry; Convex function; Unit vector; Positive semidefinite; Projection (linear algebra); Calculation; Operator algebra; Real number; Probability measure; Quantity; Equivalence relation; Invertible matrix; Cauchy matrix; Natural number; Equation; Matrix (mathematics); Operator system; Probability; Hyperbolic function; Gamma function; Infinite divisibility (probability); Unitary operator; Diagonal matrix; Coefficient; Standard basis; Matrix unit; Quantum information; Harmonic analysis; Extreme point; Commutative property; Metric space; Binomial theorem; Differential geometry; Quantum statistical mechanics; Fourier transform; Cauchy–Schwarz inequality

Zielgruppe(n) für dieses Buch

For an expert adult audience, including professional development and academic research

Positive Definite Matrices

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