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