Introduction to Matrix
            Analysis and Applications
                Fumio Hiai and D´enes Petz
              Graduate School of Information Sciences
          Tohoku University, Aoba-ku, Sendai, 980-8579, Japan
                   E-mail: fumio.hiai@gmail.com
               Alfr´ed R´enyi Institute of Mathematics
           Re´altanoda utca 13-15, H-1364 Budapest, Hungary
                   E-mail: petz.denes@renyi.mta.hu
                    Preface
                       Apart of the material of this book is based on the lectures of the authors
                    in the Graduate School of Information Sciences of Tohoku University and
                    in the Budapest University of Technology and Economics. The aim of the
                    lectures was to explain certain important topics on matrix analysis from the
                    point of view of functional analysis. The concept of Hilbert space appears
                    many times, but only finite-dimensional spaces are used. The book treats
                    some aspects of analysis related to matrices including such topics as matrix
                    monotonefunctions, matrix means, majorization, entropies, quantum Markov
                    triplets. There are several popular matrix applications for quantum theory.
                       The book is organized into seven chapters. Chapters 1-3 form an intro-
                    ductory part of the book and could be used as a textbook for an advanced
                    undergraduate special topics course. The word “matrix” started in 1848 and
                    applications appeared in many different areas. Chapters 4-7 contain a num-
                    ber of more advanced and less known topics. They could be used for an ad-
                    vanced specialized graduate-level course aimed at students who will specialize
                    in quantum information. But the best use for this part is as the reference for
                    active researchers in the field of quantum information theory. Researchers in
                    statistics, engineering and economics may also find this book useful.
                       Chapter 1 contains the basic subjects. We prefer the Hilbert space con-
                    cepts, so complex numbers are used. Spectrum and eigenvalues are impor-
                    tant. Determinant and trace are used later in several applications. The tensor
                    product has symmetric and antisymmetric subspaces. In this book “positive”
                    means ≥ 0, the word “non-negative” is not used here. The end of the chapter
                    contains many exercises.
                       Chapter 2 contains block-matrices, partial ordering and an elementary
                    theory of von Neumann algebras in finite-dimensional setting. The Hilbert
                    space concept requires the projections P = P2 = P∗. Self-adjoint matrices are
                    linear combinations of projections. Not only the single matrices are required,
                    but subalgebras are also used. The material includes Kadison’s inequality
                    and completely positive mappings.
                       Chapter 3 contains matrix functional calculus. Functional calculus pro-
                    vides a new matrix f(A) when a matrix A and a function f are given. This
                    is an essential tool in matrix theory as well as in operator theory. A typical
                                                               P
                                                          A      ∞    n
                    example is the exponential function e   = n=0A =n!. If f is sufficiently
                    smooth, then f(A) is also smooth and we have a useful Fr´echet differential
                    formula.
                       Chapter 4 contains matrix monotone functions. A real functions defined
                    on an interval is matrix monotone if A ≤ B implies f(A) ≤ f(B) for Hermi-
                      4
                      tian matrices A,B whose eigenvalues are in the domain interval. We have a
                      beautiful theory on such functions, initiated by L¨owner in 1934. A highlight
                      is integral expression of such functions. Matrix convex functions are also con-
                      sidered. Graduate students in mathematics and in information theory will
                      benefit from a single source for all of this material.
                         Chapter 5 contains matrix (operator) means for positive matrices. Matrix
                      extensions of the arithmetic mean (a+b)=2 and the harmonic mean
                                                                
                                                        −1    −1 −1
                                                       a   +b
                                                           2
                      are rather trivial, however it is non-trivial to define matrix version of the
                      geometricmean√ab. ThiswasfirstmadebyPuszandWoronowicz. Ageneral
                      theory on matrix means developed by Kubo and Ando is closely related to
                      operator monotone functions on (0,∞). There are also more complicated
                      means. The mean transformation M(A,B) := m(L ,R ) is a mean of the
                                                                           A    B
                      left-multiplication L  and the right-multiplication R     recently studied by
                                           A                                 B
                      Hiai and Kosaki. Another concept is a multivariable extension of two-variable
                      matrix means.
                         Chapter6containsmajorizationsforeigenvalues andsingular values of ma-
                      trices. Majorization is a certain order relation between two real vectors. Sec-
                      tion 6.1 recalls classical material that is available from other sources. There
                      are several famous majorizations for matrices which have strong applications
                      to matrix norm inequalities in symmetric norms. For instance, an extremely
                      useful inequality is called the Lidskii-Wielandt theorem. There are several
                      famous majorizations for matrices which have strong applications to matrix
                      norm inequalities in symmetric norms.
                         The last chapter contains topics related to quantum applications. Positive
                      matrices with trace 1 are the states in quantum theories and they are also
                      called density matrices. The relative entropy appeared in 1962 and the ma-
                      trix theory has many applications in the quantum formalism. The unknown
                      quantum states can be known from the use of positive operators F(x) when
                      PxF(x) = I. This is called POVM and there are a few mathematical re-
                      sults, but in quantum theory there are much more relevant subjects. These
                      subjects are close to the authors and there are some very recent results.
                         The authors thank several colleagues for useful communications, Professor
                      Tsuyoshi Ando had several remarks.
                                                                        Fumio Hiai and D´enes Petz
                      April, 2013