Positive Definite Matrices

By Rajendra Bhatia. 2009

227 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

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Citations in the corpus

Approximate Log-Hilbert-Schmidt Distances between Covariance Operators for Image Classification (2016, unknown): “[2, 23].”
Computing the Karcher mean of symmetric positive definite matrices (2013, methodology): “This structure makes Pn a complete Riemannian manifold with negative curvature [6] in which the distance between two positive matrices A and B turns out to be δ(A,B) := ‖ log(ABA)‖F , where the norm used is the Euclidean or Frobenius norm ‖A‖F := ( ∑ ij a 2 ij) .”
Matrix estimation by Universal Singular Value Thresholding (2012, background): “) Completing positive definite matrices with missing entries has received a lot of attention in the linear algebra literature [15, 51, 61], although most of the techniques are applicable only for relatively small matrices or when a sizable fraction of the entries are observed.”
Linearly Parameterized Bandits (2008, background): “It is a well-known result in linear algebra (see, for example, Theorem 1.3.3 in Bhatia, 2007) that if A and B are two symmetric positive definite matrices, then the block matrix A X X′ B is positive semidefinite if and only if XB−1X′ ≤ A. Applying this result to the two “equivalent” (r+1)×…”