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.

Positive Definite Matrices

"Matrix Mathematics" is a reference work for users of matrices in all branches of engineering, science, and applied mathematics. This book brings together a vast body of results on matrix theory for easy reference and immediate application. Each chapter begins with the development of relevant background theory followed by a large collection of specialized results. Hundreds of identities, inequalities, and matrix facts are stated rigorously and clearly with cross references, citations to the literature, and illuminating remarks. Twelve chapters cover all of the major topics in matrix theory: preliminaries; basic matrix properties; matrix classes and transformations; matrix polynomials and rational transfer functions; matrix decompositions; generalized inverses; Kronecker and Schur algebra; positive-semidefinite matrices; norms; functions of matrices and their derivatives; the matrix exponential and stability theory; and linear systems and control theory. A detailed list of symbols, a summary of notation and conventions, an extensive bibliography with author index, and an extensive index are provided for ease of use. The book will be useful for students at both the undergraduate and graduate levels, as well as for researchers and practitioners in all branches of engineering, science, and applied mathematics.

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Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory

Combinatorial Matrix Theory

This short tutorial presents two mathematical techniques namely Majorization Theory and Matrix-Monotone Functions, reviews their basic definitions and describes their concepts clearly with many illustrative examples. In addition to this tutorial, new results are presented with respect to Schur-convex functions and regarding the properties of matrix-monotone functions.

The techniques are applied to solve communication and information theoretic problems in wireless communications. The impact of spatial correlation in multiple antenna systems is characterized for many important performance measures, e.g., average mutual information, outage probability, error performance, minimum Eb/N0 and wide-band slope, zero-outage capacity, and capacity region. The impact of user distribution in cellular systems is characterized for different scenarios including perfectly informed transmitters and receivers, regarding, e.g., the average sum rate, the outage sum rate, maximum throughput. Finally, a unified framework for the performance analysis of multiple antenna systems is developed based on matrix-monotone functions. The optimization of transmit strategies for multiple antennas is carried out by optimization of matrix-monotone functions. The results within this framework resemble and complement the various results on optimal transmit strategies in single-user and multiple-user multiple-antenna systems.

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Majorization and Matrix Monotone Functions in Wireless Communications

We study iterative methods for finding the maximal Hermitian positive definite solutions of the matrix equations X + A * X -1 A = Q and X - A * X -1 A = Q, where Q is Hermitian positive definite. General convergence results are given for the basic fixed point iteration for both equations. Newton's method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation. Numerical results are reported to illustrate the convergence behaviour of various algorithms.

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Iterative solution of two matrix equations

An efficient and numerically stable implementation of a known algorithm is suggested for finding the extremal positive definite solutions of the matrix equation $X+A^*X^{-1}A=I$, if such solutions exist. The convergence rate is analyzed. A new algorithm that avoids matrix inversion is presented. Numerical examples are given to illustrate the effectiveness of the algorithms.

Computing the Extremal Positive Definite Solutions of a Matrix Equation

Let A,B be positive semidefinite matrices and $|||\cdot|||$ any unitarily invariant norm on the space of matrices. We show $ |||f(A) + f(B)||| \geq |||f(A + B)|||$ for any non-negative operator monotone function f(t) on $[0,\infty)$, and $ |||g(A) + g(B)||| \leq |||g(A + B)||| $ for non-negative increasing function g(t) on $[0,\infty)$ with g(0) = 0 and $g(\infty) = \infty$, whose inverse function is operator monotone.

Norm inequalities related to operator monotone functions

On the matrix equation X + ATX−1A = I

Let Mn be the space of n × n complex matrices. For $A\in M_n,$ let $s(A)\equiv (s_1(A),\dotsc,\break s_n(A)),$ where $s_1(A)\ge\cdots\ge s_n(A)$ are the singular values of A. We prove that if $A,B\in M_n$ are positive semidefinite, then (i) $s_j(A-B)\le s_j(A\oplus B), j=1,2, . . . ,n, and (ii) the weak log-majorization relations $s(A-|z|B)\prec_{wlog} s(A+zB)\prec_{wlog} s(A+|z|B)$ hold for any complex number z. This sharpens some results due to R. Bhatia and F. Kittaneh.

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Singular Values of Differences of Positive Semidefinite Matrices

We determine the exact range of the smallest and largest eigenvalues of real symmetric matrices of a given order whose entries are in a given interval. The maximizing and minimizing matrices are specified. We also consider the maximal spread of such matrices.

Xingzhi Zhan

Papers

Computing the Extremal Positive Definite Solutions of a Matrix Equation

Norm inequalities related to operator monotone functions

On the matrix equation X + ATX−1A = I

Singular Values of Differences of Positive Semidefinite Matrices

Extremal Eigenvalues of Real Symmetric Matrices with Entries in an Interval