Introduction to Matrix Analysis and Applications

1 Introduction to MatrixAnalysis and ApplicationsFumio Hiai and D enes PetzGraduate School of Information SciencesTohoku University, Aoba-ku, Sendai, 980-8579, JapanE-mail: ed R enyi Institute of MathematicsRe altanoda utca 13-15, H-1364 Budapest, HungaryE-mail: part of the material of this book is based on the lectures of the authorsin the Graduate School of Information Sciences of Tohoku University andin the Budapest University of Technology and Economics. The aim of thelectures was to explain certain important topics on Matrix Analysis from thepoint of view of functional Analysis .

2 The concept of Hilbert space appearsmany times, but only finite-dimensional spaces are used. The book treatssome aspects of Analysis related to matrices including such topics asmatrixmonotone functions, Matrix means, majorization, entropies, quantum Markovtriplets. There are several popular Matrix Applications for quantum 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 advancedundergraduate special topics course. The word Matrix started in 1848 andapplications appeared in many different areas.

3 Chapters 4-7 contain a num-ber of more advanced and less known topics. They could be used foran ad-vanced specialized graduate-level course aimed at students who will specializein quantum information. But the best use for this part is as the reference foractive researchers in the field of quantum information theory. Researchers instatistics, engineering and economics may also find this book 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 .

4 The tensorproduct has symmetric and antisymmetric subspaces. In this book positive means 0, the word non-negative is not used here. The end of the chaptercontains many 2 contains block-matrices, partial ordering and an elementarytheory of von Neumann algebras in finite-dimensional setting. The Hilbertspace concept requires the projectionsP=P2=P . Self-adjoint matrices arelinear combinations of projections. Not only the single matrices are required,but subalgebras are also used. The material includes Kadison s inequalityand completely positive 3 contains Matrix functional calculus.

5 Functional calculuspro-vides a new matrixf(A) when a matrixAand a functionfare given. Thisis an essential tool in Matrix theory as well as in operator theory. Atypicalexample is the exponential functioneA=P n=0An/n!. Iffis sufficientlysmooth, thenf(A) is also smooth and we have a useful Fr echet 4 contains Matrix monotone functions. A real functions definedon an interval is Matrix monotone ifA Bimpliesf(A) f(B) for Hermi-4tian matricesA,Bwhose eigenvalues are in the domain interval. We have abeautiful theory on such functions, initiated by L owner in 1934. Ahighlightis integral expression of such functions.

6 Matrix convex functionsare also con-sidered. Graduate students in mathematics and in information theory willbenefit from a single source for all of this 5 contains Matrix (operator) means for positive matrices. Matrixextensions of the arithmetic mean (a+b)/2 and the harmonic mean a 1+b 12 1are rather trivial, however it is non-trivial to define Matrix version of thegeometric mean ab. This was first made by Pusz and Woronowicz. A generaltheory on Matrix means developed by Kubo and Ando is closely relatedtooperator monotone functions on (0, ). There are also more complicatedmeans.

7 The mean transformationM(A,B) :=m(LA,RB) is a mean of theleft-multiplicationLAand the right-multiplicationRBrecently studied byHiai and Kosaki. Another concept is a multivariable extension of two-variablematrix 6 contains majorizations for eigenvalues and singular values of ma-trices. Majorization is a certain order relation between two real vectors. Sec-tion recalls classical material that is available from other sources. Thereare several famous majorizations for matrices which have strongapplicationsto Matrix norm inequalities in symmetric norms. For instance, an extremelyuseful inequality is called the Lidskii-Wielandt theorem.

8 There are severalfamous majorizations for matrices which have strong Applications to matrixnorm inequalities in symmetric last chapter contains topics related to quantum Applications . Positivematrices with trace 1 are the states in quantum theories and they are alsocalled density matrices. The relative entropy appeared in 1962 and the ma- trix theory has many Applications in the quantum formalism. The unknownquantum states can be known from the use of positive operatorsF(x) whenPxF(x) =I. This is called POVM and there are a few mathematical re-sults, but in quantum theory there are much more relevant subjects.

9 Thesesubjects are close to the authors and there are some very recent authors thank several colleagues for useful communications, ProfessorTsuyoshi Ando had several Hiai and D enes PetzApril, 2013 Contents1 Fundamentals of operators and Basics on matrices .. Hilbert space .. Jordan canonical form .. Spectrum and eigenvalues .. Trace and determinant .. Positivity and absolute value .. Tensor product .. Notes and remarks .. Exercises .. 492 Mappings and Block-matrices .. Partial ordering .. Projections .. Subalgebras.

10 Kernel functions .. Positivity preserving mappings .. Notes and remarks .. Exercises .. 983 Functional calculus and The exponential function .. Other functions .. Derivation .. Fr echet derivatives .. Notes and remarks .. Exercises .. 1334 Matrix monotone functions and Some examples of functions .. Convexity .. Pick functions .. L owner s theorem .. Some Applications .. Notes and remarks .. Exercises .. 1845 Matrix means and The geometric mean .. General theory .. Mean examples .. Mean transformation.