Completely Positive Matrices

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Completely Positive Matrices

Abraham Berman,Naomi Shaked-Monderer
Completely Positive Matrices Book PDF

Author : Abraham Berman,Naomi Shaked-Monderer
Publisher : World Scientific
Page : 222 pages
File Size : 42,7 Mb
Release : 2003
Category : Mathematics
ISBN : 9812795219

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Completely Positive Matrices by Abraham Berman,Naomi Shaked-Monderer Pdf

A real matrix is positive semidefinite if it can be decomposed as A = BBOC . In some applications the matrix B has to be elementwise nonnegative. If such a matrix exists, A is called completely positive. The smallest number of columns of a nonnegative matrix B such that A = BBOC is known as the cp- rank of A . This invaluable book focuses on necessary conditions and sufficient conditions for complete positivity, as well as bounds for the cp- rank. The methods are combinatorial, geometric and algebraic. The required background on nonnegative matrices, cones, graphs and Schur complements is outlined. Contents: Preliminaries: Matrix Theoretic Background; Positive Semidefinite Matrices; Nonnegative Matrices and M -Matrices; Schur Complements; Graphs; Convex Cones; The PSD Completion Problem; Complete Positivity: Definition and Basic Properties; Cones of Completely Positive Matrices; Small Matrices; Complete Positivity and the Comparison Matrix; Completely Positive Graphs; Completely Positive Matrices Whose Graphs are Not Completely Positive; Square Factorizations; Functions of Completely Positive Matrices; The CP Completion Problem; CP Rank: Definition and Basic Results; Completely Positive Matrices of a Given Rank; Completely Positive Matrices of a Given Order; When is the CP-Rank Equal to the Rank?. Readership: Upper level undergraduates, graduate students, academics and researchers interested in matrix theory."

Completely Positive Matrices

Abraham Berman,Naomi Shaked-Monderer
Completely Positive Matrices Book PDF

Author : Abraham Berman,Naomi Shaked-Monderer
Publisher : World Scientific
Page : 216 pages
File Size : 43,9 Mb
Release : 2003-04-11
Category : Mathematics
ISBN : 9789814486002

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Completely Positive Matrices by Abraham Berman,Naomi Shaked-Monderer Pdf

A real matrix is positive semidefinite if it can be decomposed as A=BB′. In some applications the matrix B has to be elementwise nonnegative. If such a matrix exists, A is called completely positive. The smallest number of columns of a nonnegative matrix B such that A=BB′ is known as the cp-rank of A. This invaluable book focuses on necessary conditions and sufficient conditions for complete positivity, as well as bounds for the cp-rank. The methods are combinatorial, geometric and algebraic. The required background on nonnegative matrices, cones, graphs and Schur complements is outlined. Contents: Preliminaries:Matrix Theoretic BackgroundPositive Semidefinite MatricesNonnegative Matrices and M-MatricesSchur ComplementsGraphsConvex ConesThe PSD Completion ProblemComplete Positivity:Definition and Basic PropertiesCones of Completely Positive MatricesSmall MatricesComplete Positivity and the Comparison MatrixCompletely Positive GraphsCompletely Positive Matrices Whose Graphs are Not Completely PositiveSquare FactorizationsFunctions of Completely Positive MatricesThe CP Completion ProblemCP Rank:Definition and Basic ResultsCompletely Positive Matrices of a Given RankCompletely Positive Matrices of a Given OrderWhen is the CP-Rank Equal to the Rank? Readership: Upper level undergraduates, graduate students, academics and researchers interested in matrix theory. Keywords:Reviews:“Overall, this appears to be a highly delightful book to read, study, and teach from.”Zentralblatt MATH “The topics are of interest mainly from an applied mathematician's point of view, but the techniques and the difficulties make them appealing for the pure mathematician as well.”Mathematical Reviews

Copositive And Completely Positive Matrices

Naomi Shaked-monderer,Abraham Berman
Copositive And Completely Positive Matrices Book PDF

Author : Naomi Shaked-monderer,Abraham Berman
Publisher : World Scientific
Page : 562 pages
File Size : 47,9 Mb
Release : 2021-02-09
Category : Mathematics
ISBN : 9789811204364

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Copositive And Completely Positive Matrices by Naomi Shaked-monderer,Abraham Berman Pdf

This book is an updated and extended version of Completely Positive Matrices (Abraham Berman and Naomi Shaked-Monderer, World Scientific 2003). It contains new sections on the cone of copositive matrices, which is the dual of the cone of completely positive matrices, and new results on both copositive matrices and completely positive matrices.The book is an up to date comprehensive resource for researchers in Matrix Theory and Optimization. It can also serve as a textbook for an advanced undergraduate or graduate course.

Positive Linear Maps of Operator Algebras

Erling Størmer
Positive Linear Maps of Operator Algebras Book PDF

Author : Erling Størmer
Publisher : Springer Science & Business Media
Page : 135 pages
File Size : 55,9 Mb
Release : 2012-12-13
Category : Mathematics
ISBN : 9783642343698

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Positive Linear Maps of Operator Algebras by Erling Størmer Pdf

This volume, setting out the theory of positive maps as it stands today, reflects the rapid growth in this area of mathematics since it was recognized in the 1990s that these applications of C*-algebras are crucial to the study of entanglement in quantum theory. The author, a leading authority on the subject, sets out numerous results previously unpublished in book form. In addition to outlining the properties and structures of positive linear maps of operator algebras into the bounded operators on a Hilbert space, he guides readers through proofs of the Stinespring theorem and its applications to inequalities for positive maps. The text examines the maps’ positivity properties, as well as their associated linear functionals together with their density operators. It features special sections on extremal positive maps and Choi matrices. In sum, this is a vital publication that covers a full spectrum of matters relating to positive linear maps, of which a large proportion is relevant and applicable to today’s quantum information theory. The latter sections of the book present the material in finite dimensions, while the text as a whole appeals to a wider and more general readership by keeping the mathematics as elementary as possible throughout.

Positive Definite Matrices

Rajendra Bhatia
Positive Definite Matrices Book PDF

Author : Rajendra Bhatia
Publisher : Princeton University Press
Page : 264 pages
File Size : 51,7 Mb
Release : 2015-09-01
Category : Mathematics
ISBN : 9780691168258

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Positive Definite Matrices by Rajendra Bhatia Pdf

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.

Completely Bounded Maps and Operator Algebras

Vern Paulsen
Completely Bounded Maps and Operator Algebras Book PDF

Author : Vern Paulsen
Publisher : Cambridge University Press
Page : 316 pages
File Size : 43,5 Mb
Release : 2002
Category : Mathematics
ISBN : 0521816696

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Completely Bounded Maps and Operator Algebras by Vern Paulsen Pdf

Table of contents

Totally Positive Matrices

Allan Pinkus
Totally Positive Matrices Book PDF

Author : Allan Pinkus
Publisher : Unknown
Page : 182 pages
File Size : 45,5 Mb
Release : 2010
Category : Electronic books
ISBN : 051169038X

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Totally Positive Matrices by Allan Pinkus Pdf

"Totally positive matrices constitute a particular class of matrices, the study of which was initiated by analysts because of its many applications in diverse areas. This modern account of the subject is comprehensive and thorough, with careful treatment of the central properties of totally positive matrices, full proofs and a complete bibliography. The history of the subject is also described: in particular, the book ends with a tribute to the four people who have made the most notable contributions to the history of total positivity: I.J. Schoenberg, M.G. Krein, F.R. Gantmacher and S. Karlin. This monograph will appeal to those with an interest in matrix theory, to those who use or have used total positivity, and to anyone who wishes to learn about this rich and interesting subject"--Provided by publisher.

Tensor Analysis

Liqun Qi,Ziyan Luo
Tensor Analysis Book PDF

Author : Liqun Qi,Ziyan Luo
Publisher : SIAM
Page : 313 pages
File Size : 55,8 Mb
Release : 2017-04-19
Category : Mathematics
ISBN : 9781611974744

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Tensor Analysis by Liqun Qi,Ziyan Luo Pdf

Tensors, or hypermatrices, are multi-arrays with more than two indices. In the last decade or so, many concepts and results in matrix theory?some of which are nontrivial?have been extended to tensors and have a wide range of applications (for example, spectral hypergraph theory, higher order Markov chains, polynomial optimization, magnetic resonance imaging, automatic control, and quantum entanglement problems). The authors provide a comprehensive discussion of this new theory of tensors. Tensor Analysis: Spectral Theory and Special Tensors is unique in that it is the first book on these three subject areas: spectral theory of tensors; the theory of special tensors, including nonnegative tensors, positive semidefinite tensors, completely positive tensors, and copositive tensors; and the spectral hypergraph theory via tensors. ?

Graphs and Matrices

Ravindra B. Bapat
Graphs and Matrices Book PDF

Author : Ravindra B. Bapat
Publisher : Springer
Page : 193 pages
File Size : 45,6 Mb
Release : 2014-09-19
Category : Mathematics
ISBN : 9781447165699

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Graphs and Matrices by Ravindra B. Bapat Pdf

This new edition illustrates the power of linear algebra in the study of graphs. The emphasis on matrix techniques is greater than in other texts on algebraic graph theory. Important matrices associated with graphs (for example, incidence, adjacency and Laplacian matrices) are treated in detail. Presenting a useful overview of selected topics in algebraic graph theory, early chapters of the text focus on regular graphs, algebraic connectivity, the distance matrix of a tree, and its generalized version for arbitrary graphs, known as the resistance matrix. Coverage of later topics include Laplacian eigenvalues of threshold graphs, the positive definite completion problem and matrix games based on a graph. Such an extensive coverage of the subject area provides a welcome prompt for further exploration. The inclusion of exercises enables practical learning throughout the book. In the new edition, a new chapter is added on the line graph of a tree, while some results in Chapter 6 on Perron-Frobenius theory are reorganized. Whilst this book will be invaluable to students and researchers in graph theory and combinatorial matrix theory, it will also benefit readers in the sciences and engineering.

Introduction to Matrix Analysis and Applications

Fumio Hiai,Dénes Petz
Introduction to Matrix Analysis and Applications Book PDF

Author : Fumio Hiai,Dénes Petz
Publisher : Springer Science & Business Media
Page : 337 pages
File Size : 40,5 Mb
Release : 2014-02-06
Category : Mathematics
ISBN : 9783319041506

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Introduction to Matrix Analysis and Applications by Fumio Hiai,Dénes Petz Pdf

Matrices can be studied in different ways. They are a linear algebraic structure and have a topological/analytical aspect (for example, the normed space of matrices) and they also carry an order structure that is induced by positive semidefinite matrices. The interplay of these closely related structures is an essential feature of matrix analysis. This book explains these aspects of matrix analysis from a functional analysis point of view. After an introduction to matrices and functional analysis, it covers more advanced topics such as matrix monotone functions, matrix means, majorization and entropies. Several applications to quantum information are also included. Introduction to Matrix Analysis and Applications is appropriate for an advanced graduate course on matrix analysis, particularly aimed at studying quantum information. It can also be used as a reference for researchers in quantum information, statistics, engineering and economics.

Nonnegative Matrices and Applications

R. B. Bapat,T. E. S. Raghavan
Nonnegative Matrices and Applications Book PDF

Author : R. B. Bapat,T. E. S. Raghavan
Publisher : Cambridge University Press
Page : 351 pages
File Size : 50,8 Mb
Release : 1997-03-28
Category : Mathematics
ISBN : 9780521571678

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Nonnegative Matrices and Applications by R. B. Bapat,T. E. S. Raghavan Pdf

This book provides an integrated treatment of the theory of nonnegative matrices (matrices with only positive numbers or zero as entries) and some related classes of positive matrices, concentrating on connections with game theory, combinatorics, inequalities, optimisation and mathematical economics. The wide variety of applications, which include price fixing, scheduling and the fair division problem, have been carefully chosen both for their elegant mathematical content and for their accessibility to students with minimal preparation. Many results in matrix theory are also presented. The treatment is rigorous and almost all results are proved completely. These results and applications will be of great interest to researchers in linear programming, statistics and operations research. The minimal prerequisites also make the book accessible to first-year graduate students.

Introduction to Applied Linear Algebra

Stephen Boyd,Lieven Vandenberghe
Introduction to Applied Linear Algebra Book PDF

Author : Stephen Boyd,Lieven Vandenberghe
Publisher : Cambridge University Press
Page : 477 pages
File Size : 52,7 Mb
Release : 2018-06-07
Category : Business & Economics
ISBN : 9781316518960

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Introduction to Applied Linear Algebra by Stephen Boyd,Lieven Vandenberghe Pdf

A groundbreaking introduction to vectors, matrices, and least squares for engineering applications, offering a wealth of practical examples.

The Theory of Quantum Information

John Watrous
The Theory of Quantum Information Book PDF

Author : John Watrous
Publisher : Unknown
Page : 599 pages
File Size : 50,5 Mb
Release : 2018-04-26
Category : Computers
ISBN : 9781107180567

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The Theory of Quantum Information by John Watrous Pdf

Formal development of the mathematical theory of quantum information with clear proofs and exercises. For graduate students and researchers.

Totally Nonnegative Matrices

Shaun M. Fallat,Charles R. Johnson
Totally Nonnegative Matrices Book PDF

Author : Shaun M. Fallat,Charles R. Johnson
Publisher : Princeton University Press
Page : 264 pages
File Size : 42,6 Mb
Release : 2022-12-13
Category : Mathematics
ISBN : 9780691242415

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Totally Nonnegative Matrices by Shaun M. Fallat,Charles R. Johnson Pdf

Totally nonnegative matrices arise in a remarkable variety of mathematical applications. This book is a comprehensive and self-contained study of the essential theory of totally nonnegative matrices, defined by the nonnegativity of all subdeterminants. It explores methodological background, historical highlights of key ideas, and specialized topics. The book uses classical and ad hoc tools, but a unifying theme is the elementary bidiagonal factorization, which has emerged as the single most important tool for this particular class of matrices. Recent work has shown that bidiagonal factorizations may be viewed in a succinct combinatorial way, leading to many deep insights. Despite slow development, bidiagonal factorizations, along with determinants, now provide the dominant methodology for understanding total nonnegativity. The remainder of the book treats important topics, such as recognition of totally nonnegative or totally positive matrices, variation diminution, spectral properties, determinantal inequalities, Hadamard products, and completion problems associated with totally nonnegative or totally positive matrices. The book also contains sample applications, an up-to-date bibliography, a glossary of all symbols used, an index, and related references.

Matrix Analysis

Roger A. Horn,Charles R. Johnson
Matrix Analysis Book PDF

Author : Roger A. Horn,Charles R. Johnson
Publisher : Cambridge University Press
Page : 662 pages
File Size : 55,5 Mb
Release : 2012-10-22
Category : Mathematics
ISBN : 0521548233

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Matrix Analysis by Roger A. Horn,Charles R. Johnson Pdf

Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of the acclaimed text presents results of both classic and recent matrix analysis using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications. The authors have thoroughly revised, updated, and expanded on the first edition. The book opens with an extended summary of useful concepts and facts and includes numerous new topics and features, such as: - New sections on the singular value and CS decompositions - New applications of the Jordan canonical form - A new section on the Weyr canonical form - Expanded treatments of inverse problems and of block matrices - A central role for the Von Neumann trace theorem - A new appendix with a modern list of canonical forms for a pair of Hermitian matrices and for a symmetric-skew symmetric pair - Expanded index with more than 3,500 entries for easy reference - More than 1,100 problems and exercises, many with hints, to reinforce understanding and develop auxiliary themes such as finite-dimensional quantum systems, the compound and adjugate matrices, and the Loewner ellipsoid - A new appendix provides a collection of problem-solving hints.