Positive Definite Matrices

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.

[1]  Jimmie D. Lawson,et al.  The Geometric Mean, Matrices, Metrics, and More , 2001, Am. Math. Mon..

[2]  Aarnout Brombacher,et al.  Probability... , 2009, Qual. Reliab. Eng. Int..

[3]  Rajendra Bhatia,et al.  Comparing a Matrix to its Off-Diagonal Part , 1989 .

[4]  G. Lindblad.  Expectations and entropy inequalities for finite quantum systems , 1974 .

[5]  W. Pusz,et al.  Functional calculus for sesquilinear forms and the purification map , 1975 .

[6]  D. Petz,et al.  Quantum Entropy and Its Use , 1993 .

[7]  L. Schwartz.  Théorie des distributions , 1966 .

[8]  P. Halmos.  A Hilbert Space Problem Book , 1967 .

[9]  J. Joseph,et al.  Fourier transforms , 2012 .

[10]  A. Wehrl.  General properties of entropy , 1978 .

[11]  Man-Duen Choi.  A schwarz inequality for positive linear maps on $C^{\ast}$-algebras , 1974 .

[12]  R. Bhatia,et al.  Norm Inequalities for Positive Operators , 1998 .

[13]  Karl Löwner.  Über monotone Matrixfunktionen , 1934 .

[14]  Jim Hefferon,et al.  Linear Algebra , 2012 .

[15]  R. Mathias.  An arithmetic-geometric-harmonic mean inequality involving Hadamard products , 1993 .

[16]  E. Lieb,et al.  Some operator inequalities of the schwarz type , 1974 .

[17]  F. Hiai,et al.  Means for Matrices and Comparison of their Norms , 1999 .

[18]  E. Lieb,et al.  Proof of the strong subadditivity of quantum‐mechanical entropy , 1973 .

[19]  R. Bhatia.  On the exponential metric increasing property , 2003 .

[20]  V. Paulsen.  Completely bounded maps and dilations , 1987 .

[21]  Irving Segal,et al.  Notes towards the construction of nonlinear relativistic quantum fields. III: Properties of the $C^*$-dynamics for a certain class of interactions , 1969 .

[22]  T. Andô,et al.  On some operator inequalities , 1987 .

[23]  T. Andô,et al.  Means of positive linear operators , 1980 .

[24]  R. Bhatia.  Interpolating the arithmetic-geometric mean inequality and its operator version , 2006 .

[25]  V. Müller.  The numerical radius of a commuting product. , 1988 .

[26]  R. Bhatia,et al.  Variation of induced linear operators , 2002 .

[27]  F. Hiai,et al.  The Golden-Thompson trace inequality is complemented , 1993 .

[28]  P. Meyer,et al.  Quantum Probability for Probabilists , 1993 .

[29]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[30]  Chandler Davis,et al.  More matrix forms of the arithmetic-geometric mean inequality , 1993 .

[31]  R. Horn.  ON BOUNDARY VALUES OF A SCHLICHT MAPPING , 1967 .

[32]  J. Stewart.  Positive definite functions and generalizations, an historical survey , 1976 .

[33]  Fumio Hiai,et al.  Log majorization and complementary Golden-Thompson type inequalities , 1994 .

[34]  Shmuel Friedland,et al.  Variation of Grassman powers and spectra , 1981 .

[35]  Zoltán Sasvári,et al.  Positive definite and definitizable functions , 1994 .

[36]  S. Woronowicz.  Positive maps of low dimensional matrix algebras , 1976 .

[37]  M. Kwong.  Some results on matrix monotone functions , 1989 .

[38]  Kenneth R. Davidson,et al.  Numerical radii of zero-one matrices. , 1988 .

[39]  Maher Moakher.  On the Averaging of Symmetric Positive-Definite Tensors , 2006 .

[40]  E. Størmer.  Positive linear maps of Cu * -algebras , 1974 .

[41]  安藤 毅,et al.  Operator-Theoretic methods for matrix inequalities , 1998 .

[42]  J. Steele.  The Cauchy–Schwarz Master Class: References , 2004 .

[43]  Chandler Davis,et al.  More Operator Versions of the Schwarz Inequality , 2000 .

[44]  Roger Cuppens,et al.  Decomposition of multivariate probabilities , 1975 .

[45]  S. Golden.  LOWER BOUNDS FOR THE HELMHOLTZ FUNCTION , 1965 .

[46]  R. Bhatia,et al.  On the singular values of a product of operators , 1990 .

[47]  Matrix Operations Induced by Network Connections , 1975 .

[48]  Charles R. Johnson,et al.  Eigenvalues of Words in Two Positive Definite Letters , 2002, SIAM J. Matrix Anal. Appl..

[49]  安藤 毅.  Topics on operator inequalities , 1978 .

[50]  M. Bridson,et al.  Metric Spaces of Non-Positive Curvature , 1999 .

[51]  C. Davis.  Notions generalizing convexity for functions defined on spaces of matrices , 1963 .

[52]  Mei Han An,et al.  accuracy and stability of numerical algorithms , 1991 .

[53]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[54]  Rajendra Bhatia,et al.  Positive Definite Functions and Operator Inequalities , 2000 .

[55]  R. Bhatia,et al.  Notes on matrix arithmetic–geometric mean inequalities , 2000 .

[56]  E. Cartan.  Groupes simples clos et ouverts et géométrie riemannienne , 1929 .

[57]  E. Haynsworth.  Determination of the inertia of a partitioned Hermitian matrix , 1968 .

[58]  S. Bochner,et al.  Vorlesungen über Fouriersche Integrale , 1952 .

[59]  D. Bessis,et al.  Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics , 1975 .

[60]  M. Gromov,et al.  Manifolds of Nonpositive Curvature , 1985 .

[61]  Erhard Heinz,et al.  Beiträge zur Störungstheorie der Spektralzerleung , 1951 .

[62]  R. Kadison.  A Generalized Schwarz Inequality and Algebraic Invariants for Operator Algebras , 1952 .

[63]  W. Ip,et al.  A matrix version of the Wielandt inequality and its applications to statistics , 1999 .

[64]  S. Lang.  Fundamentals of differential geometry , 1998 .

[65]  M. Fiedler,et al.  A new positive definite geometric mean of two positive definite matrices , 1997 .

[66]  T. Andô,et al.  Norm inequalities related to operator monotone functions , 1999 .

[67]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[68]  G. Lindblad.  Completely positive maps and entropy inequalities , 1975 .

[69]  E. Lukács.  CERTAIN ENTIRE CHARACTERISTIC FUNCTIONS , 2005 .

[70]  Rajendra Bhatia,et al.  Infinitely Divisible Matrices , 2006, Am. Math. Mon..

[71]  J. Schur,et al.  Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind. , 1917 .

[72]  R. Subramanian,et al.  Inequalities between means of positive operators , 1978, Mathematical Proceedings of the Cambridge Philosophical Society.

[73]  Elliott H. Lieb,et al.  Equivalent Forms of the Bessis–Moussa–Villani Conjecture , 2004 .

[74]  Rajendra Bhatia,et al.  Positive Linear Maps and the Lyapunov Equation , 2002 .

[75]  R. Bhatia,et al.  Frechet derivatives of the power function , 2000 .

[76]  H. Kosaki.  On infinite divisibility of positive definite functions arising from operator means , 2008 .

[77]  T. Raghavan,et al.  Nonnegative Matrices and Applications , 1997 .

[78]  J. Schur.  Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen. , 1911 .

[79]  M. Fiedler,et al.  Matrix Inequalities , 1966 .

[80]  Man-Duen Choi.  Completely positive linear maps on complex matrices , 1975 .

[81]  M. Redondo,et al.  α-Derivations , 1995, Canadian Mathematical Bulletin.

[82]  Rajendra Bhatia,et al.  A Better Bound on the Variance , 2000, Am. Math. Mon..

[83]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[84]  K. Kraus.  General state changes in quantum theory , 1971 .

[85]  R. Bhatia,et al.  Variation of real powers of positive operators , 1994 .

[86]  David W. Lewis,et al.  Matrix theory , 1991 .

[87]  R. Bhatia,et al.  Riemannian geometry and matrix geometric means , 2006 .

[88]  E. Lieb,et al.  A Guide to Entropy and the Second Law of Thermodynamics , 1998, math-ph/9805005.

[89]  Rajendra Bhatia,et al.  Partial traces and entropy inequalities , 2003 .

[90]  Danko R. Jocić.  Cauchy–Schwarz norm inequalities for weak∗-integrals of operator valued functions , 2005 .

[91]  P. Moussa.  On the Representation of Tr(e (A-λB) ) as a Laplace Transform , 2000 .

[92]  P. Levy.  Théorie de l'addition des variables aléatoires , 1955 .

[93]  C. Herz.  Fonctions opérant sur les fonctions définies-positives , 1963 .

[94]  C. Ballantine.  On the Hadamard product , 1968 .

[95]  Horacio Porta,et al.  Convexity of the geodesic distance on spaces of positive operators , 1994 .

[96]  Simo Puntanen,et al.  Some comments on several matrix inequalities with applications to canonical correlations: Historical background and recent developments , 2002 .

[97]  G. Pólya,et al.  Remarks on Characteristic Functions , 1949 .

[98]  P. Shor.  Equivalence of Additivity Questions in Quantum Information Theory , 2004 .

[99]  Elliott H. Lieb,et al.  The Mathematical Structure of the Second Law of Thermodynamics , 2001 .

[100]  J. Bunce.  Shorted operators and the structure of operators with numerical radius one , 1988 .

[101]  R. Bhatia,et al.  Noncommutative geometric means , 2006 .

[102]  T. Andô.  Concavity of certain maps on positive definite matrices and applications to Hadamard products , 1979 .

[103]  M. Marcus,et al.  Partitioned hermitian matrices , 1971 .

[104]  Demetrio Stojanoff,et al.  Geometric operator inequalities , 1997 .

[105]  F. Hiai,et al.  Comparison of Various Means for Operators , 1999 .

[106]  Iaira Boissevain.  [Pinching]. , 2004, Tijdschrift voor diergeneeskunde.

[107]  R. Mathias.  A note on "More Operator Versions of the Schwarz Inequality" , 2004 .

[108]  Danko R. Jocić.  Cauchy-Schwarz and means inequalities for elementary operators into norm ideals , 1998 .

[109]  L. Hesselholt,et al.  Higher Algebra , 1937, Nature.

[110]  C. Richard Johnson,et al.  Matrix Completion Problems: A Survey , 1990 .

[111]  H. Kosaki.  Arithmetic–Geometric Mean and Related Inequalities for Operators , 1998 .

[112]  Gene H. Golub,et al.  Matrix computations , 1983 .

[113]  C. Berg,et al.  Harmonic Analysis on Semigroups , 1984 .

[114]  E. Størmer.  Positive linear maps of operator algebras , 2012 .

[115]  V. Paulsen.  Completely Bounded Maps and Operator Algebras: Contents , 2003 .

[116]  B. Russo,et al.  A note on unitary operators in $C\sp{\ast}$-algebras , 1966 .

[117]  Roger A. Horn,et al.  Norm bounds for Hadamard products and an arithmetic - geometric mean inequality for unitarily invariant norms , 1995 .

[118]  R. Jackson.  Inequalities , 2007, Algebra for Parents.

[119]  Horacio Porta,et al.  GEODESICS AND OPERATOR MEANS IN THE SPACE OF POSITIVE OPERATORS , 1993 .

[120]  R. Bhatia,et al.  Differentiation of Operator Functions and Perturbation Bounds , 1998 .

[121]  T. Andô,et al.  Non-Linear Completely Positive Maps , 1986 .

[122]  H.,et al.  Convex Trace Functions and the Wigner-Yanase-Dyson Conjecture , 2022 .

[123]  R. Bhatia.  Variation of symmetric tensor powers and permanents , 1984 .

[124]  C. Thompson.  Inequality with Applications in Statistical Mechanics , 1965 .

[125]  T. Andô,et al.  Induced norms of the Schur multiplier operator , 1991 .

[126]  R. Merris.  Trace functions, I , 1971 .

[127]  W. Stinespring.  Positive functions on *-algebras , 1955 .

[128]  R. Bhatia.  First and second order perturbation bounds for the operator absolute value , 1994 .

[129]  Dénes Petz,et al.  Means of Positive Numbers and Matrices , 2005, SIAM J. Matrix Anal. Appl..

[130]  P. Lancaster,et al.  The Algebraic Riccati Equation , 1995 .

[131]  Rajendra Bhatia,et al.  Generalized Lyapunov Equations and Positive Definite Functions , 2005, SIAM J. Matrix Anal. Appl..

[132]  R. Duffin,et al.  Series and parallel addition of matrices , 1969 .

[133]  Chandler Davis,et al.  A Schwarz inequality for convex operator functions , 1957 .

[134]  J. Neumann.  Thermodynamik quantenmechanischer Gesamtheiten , 1927 .

[135]  K. Parthasarathy.  An Introduction to Quantum Stochastic Calculus , 1992 .

[136]  V. Sunder.  A noncommutative analogue of |D(Xk|=|kXk−1| , 1982 .

[137]  W. Arveson.  On subalgebras of $C^*$-algebras , 1969 .

[138]  Fuzhen Zhang.  Matrix Theory: Basic Results and Techniques , 1999 .

[139]  B. Simon.  Trace ideals and their applications , 1979 .

[140]  F. Hiai,et al.  Means of Hilbert Space Operators , 2003 .

[141]  K. Parthasarathy,et al.  Limit theorems for uniformly infinitesimal families of positive definite kernels , 1972 .

[142]  Maher Moakher,et al.  A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices , 2005, SIAM J. Matrix Anal. Appl..

[143]  H. Gurney.  Heat Transmission , 1909, Nature.

[144]  K. Chung,et al.  Limit Distributions for Sums of Independent Random Variables. , 1955 .

[145]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[146]  Vern I. Paulsen,et al.  Schur Products and Matrix Completions , 1989 .

[147]  A. Terras.  Harmonic Analysis on Symmetric Spaces and Applications I , 1985 .

[148]  Michel Loève,et al.  Probability Theory I , 1977 .

[149]  Man Kam Kwong,et al.  On the definiteness of the solutions of certain matrix equations , 1988 .

[150]  E. Lieb,et al.  A Fundamental Property of Quantum-Mechanical Entropy , 1973 .

[151]  Gilles Pisier,et al.  Introduction to Operator Space Theory , 2003 .

[152]  A. Wehrl.  The many facets of entropy , 1991 .

[153]  N. Hitchin.  A panoramic view of riemannian geometry , 2006 .

[154]  Chi-Kwong Li.  Geometric Means , 2003 .

[155]  J. Pillis,et al.  Transformations on partitioned matrices , 1969 .