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Positive Definite Matrices

Positive Definite Matrices

Rajendra Bhatia
Copyright Date: 2007
Pages: 264
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  • Book Info
    Positive Definite Matrices
    Book Description:

    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 Matricesis 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.

    eISBN: 978-1-4008-2778-7
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-vi)
  3. Preface
    (pp. vii-x)
  4. Chapter One Positive Matrices
    (pp. 1-34)

    We begin with a quick review of some of the basic properties of positive matrices. This will serve as a warmup and orient the reader to the line of thinking followed through the book.

    Let$\mathcal{H}$be the$n$-dimensional Hilbert space${\mathbb{C}^n}$. The inner product between two vectors$x$and$y$is written as$\langle x,y\rangle $or as$x*y$. We adopt the convention that the inner product is conjugate linear in the first variable and linear in the second. We denote by$\mathcal{L}(\mathcal{H})$the space of all linear operators on$\mathcal{H}$and by${\mathbb{M}_n}(\mathbb{C})$or simply${\mathbb{M}_n}$the space of$n \times n$...

  5. Chapter Two Positive Linear Maps
    (pp. 35-64)

    In this chapter we study linear maps on spaces of matrices. We use the symbol$\Phi $for a linear map from${\mathbb{M}_n}$into${\mathbb{M}_k}$. When$k = 1$such a map is called a linear functional, and we use the lower-case symbol$\varphi $for it. The norm of$\Phi $is

    $\left\| \Phi \right\| = \mathop {\sup }\limits_{\left\| A \right\| = 1} \left\| {\Phi (A)} \right\| = \mathop {\sup }\limits_{\left\| A \right\| \leqslant 1} \left\| {\Phi (A)} \right\|$.

    In general, it is not easy to calculate this. One of the principal results of this chapter is that if$\Phi $carries positive elements of${\mathbb{M}_n}$to positive elements of${\mathbb{M}_k}$, then$\left\| \Phi \right\| = \left\| {\Phi (I)} \right\|$.

    The interplay between algebraic properties of linear maps$\Phi $and their metric properties is...

  6. Chapter Three Completely Positive Maps
    (pp. 65-100)

    For several reasons a special class of positive maps, called completely positive maps, is especially important. In Section 3.1 we study the basic properties of this class of maps. In Section 3.3 we derive some Schwarz type inequalities for this class; these are not always true for all positive maps. In Sections 3.4 and 3.5 we use general results on completely positive maps to study some important problems for matrix norms.

    Let${\mathbb{M}_m}({\mathbb{M}_n})$be the space of$m \times m$block matrices$[[{A_{ij}}]]$whose$i,j$entry is an element of${\mathbb{M}_n} = {\mathbb{M}_n}(\mathbb{C})$. Each linear map$\Phi :{\mathbb{M}_n} \to {\mathbb{M}_k}$...

  7. Chapter Four Matrix Means
    (pp. 101-140)

    Let$a$and$b$be positive numbers. Their arithmetic, geometric, and harmonic means are the familiar objects

    ${\rm{A (a, b) = }}\frac{{{\rm{a + b}}}}{{\rm{2}}}$,

    ${\rm{G(a, b) = }}\sqrt {{\rm{ab}}} $,

    ${\rm{H (a, b) = }}{\left( {\frac{{{{\rm{a}}^{{\rm{ - 1}}}}{\rm{ + }}{{\rm{b}}^{{\rm{ - 1}}}}}}{2}} \right)^{{\rm{ - 1}}}}$.

    These have several properties that any object that is called amean$M(a,b)$should have. Some of these properties are

    (i)$M(a,b) > 0$,

    (ii) If$a \leqslant b$, then$a \leqslant M(a,b) \leqslant b$,

    (iii)$M(a,b) = M(b,a)$(symmetry),

    (iv)$M(a,b)$is monotone increasing in$a,b$,

    (v)$M(\alpha a,\alpha b) = \alpha M(a,b)$for all positive numbers$a,b$, and$\alpha $,

    (vi)$M(a,b)$is continuous in$a,b$....

  8. Chapter Five Positive Definite Functions
    (pp. 141-200)

    Positive definite functions arise naturally in many areas of mathematics. In this chapter we study some of their basic properties, construct some examples, and use them to derive interesting results about positive matrices.

    Positive definite sequences were introduced in Section 1.1.3. We repeat the definition. A (doubly infinite) sequence of complex numbers$\{ {a_n}:n \in \mathbb{Z}\} $is said to bepositive definiteif for every positive integer$N$, we have

    $\sum\limits_{r,s = 0}^{N - 1} {{a_{r - s}}} {\xi _r}{\overline \xi _s} \ge 0$, (5.1)

    for every finite sequence of complex numbers${\xi _0},{\xi _1},...,{\xi _{N - 1}}$. This condition is equivalent to the requirement that for each$N = 1,2,...,$\...

  9. Chapter Six Geometry of Positive Matrices
    (pp. 201-236)

    The set of$n \times n$positive matrices is a differentiable manifold with a natural Riemannian structure. The geometry of this manifold is intimately connected with some matrix inequalities. In this chapter we explore this connection. Among other things, this leads to a deeper understanding of the geometric mean of positive matrices.

    The space${\mathbb{M}_n}$is a Hilbert space with the inner product$\langle A,B\rangle = {\text{tr }}A*B$and the associated norm${\left\| A \right\|_2} = {({\text{tr }}A*A)^{1/2}}$. The set of Hermitian matrices constitutes a real vector space${\mathbb{H}_n}$in${\mathbb{M}_n}$. The subset${\mathbb{P}_n}$consisting ofstrictlypositive matrices is an open...

  10. Bibliography
    (pp. 237-246)
  11. Index
    (pp. 247-252)
  12. Notation
    (pp. 253-254)