Positive definite matrices

Positive Definite Matrices

Volumn , Issue , 2009, Pages

  Bhatia, Rajendra ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84883937488     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Book

References (180)

1. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Dover, 1993 (reprint of original edition).
2. W. N. Anderson and R. J. Duffin, Series and parallel addition of matrices, J. Math. Anal. Appl., 26 (1969) 576-594.
3. W. N. Anderson and G. E. Trapp, Matrix operations induced by electrical network connections-a survey, in Constructive Approaches to Mathematical Models, Academic Press, 1979, pp. 53-73.
4. W. N. Anderson and G. E. Trapp, Operator means and electrical networks, in Proc. 1980 IEEE International Symposium on Circuits and Systems.
5. T. Ando, Structure of operators with numerical radius one, Acta Sci. Math. (Szeged), 34 (1973) 11-15.
6. T. Ando, Topics on Operator Inequalities, Hokkaido University, Sapporo, 1978.
7. T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl., 26 (1979) 203-241.
8. T. Ando, On some operator inequalities, Math. Ann., 279 (1987) 157-159.
9. T. Ando, Operator-Theoretic Methods for Matrix Inequalities, Sapporo, 1998.
10. T. Ando and M.-D. Choi, Non-linear completely positive maps, in Aspects of Positivity in Functional Analysis, North Holland Mathematical Studies Vol. 122, 1986, pp.3-13.
11. T. Ando and F. Hiai, Log majorization and complementary Golden-Thompson type inequalities, Linear Algebra Appl., 197/198 (1994) 113-131.
12. T. Ando, C.-K Li, and R. Mathias, Geometric Means, Linear Algebra Appl., 385 (2004) 305-334.
13. T. Ando and K. Okubo, Induced norms of the Schur multiplier operator, Linear Algebra Appl., 147 (1991) 181-199.
14. T. Ando and X. Zhan, Norm inequalities related to operator monotone functions, Math. Ann., 315 (1999) 771-780.
15. E. Andruchow, G. Corach, and D. Stojanoff, Geometric operator inequalities, Linear Algebra Appl., 258 (1997) 295-310.
16. W. B. Arveson, Subalgebras of C*-algebras, I, II, Acta Math. 123 (1969) 141-224.
17. W. B. Arveson, Subalgebras of C*-algebras, I, II, Acta Math. 128 (1972) 271-308.
18. W. B. Arveson, Nonlinear states on C*-algebras, in Operator Algebras and Mathematical Physics, Contemporary Mathematics Vol. 62, American Math. Society, 1987, pp 283-343.
19. W. Ballmann, M. Gromov, and V. Schroeder, Manifolds of Nonpositive Curvature, Birkḧauser, 1985.
20. R. B. Bapat and T.E.S. Raghavan, Nonnegative Matrices and Applications, Cambridge University Press, 1997.
21. C. Berg, J.P.R. Christensen, and P. Ressel, Harmonic Analysis on Semigroups, Springer, 1984.
22. M. Berger, A Panoramic View of Riemannian Geometry, Springer, 2003.
23. D. Bessis, P. Moussa, and M. Villani, Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics, J. Math. Phys., 16 (1975) 2318-2325.
24. K. V. Bhagwat and R. Subramanian, Inequalities between means of positive operators, Math. Proc. Cambridge Philos. Soc., 83 (1978) 393-401.
25. R. Bhatia, Variation of symmetric tensor powers and permanents, Linear Algebra Appl., 62 (1984) 269-276.
26. R. Bhatia, First and second order perturbation bounds for the operator absolute value, Linear Algebra Appl., 208/209 (1994) 367-376.
27. R. Bhatia, Matrix Analysis, Springer, 1997.
28. R. Bhatia, Pinching, trimming, truncating and averaging of matrices, Am. Math. Monthly, 107 (2000) 602-608.
29. R. Bhatia, Partial traces and entropy inequalities, Linear Algebra Appl., 375 (2003) 211-220.
30. R. Bhatia, On the exponential metric increasing property, Linear Algebra Appl., 375 (2003) 211-220.
31. R. Bhatia, Infinitely divisible matrices, Am. Math. Monthly, 113 (2006) 221-235.
32. R. Bhatia, Interpolating the arithmetic-geometric mean inequality and its operator version, Linear Algebra Appl., 413 (2006) 355-363.
33. R. Bhatia, M.-D. Choi, and C. Davis, Comparing a matrix to its off-diagonal part, Oper. Theory: Adv. Appl., 40 (1989) 151-164.
34. R. Bhatia and C. Davis, More matrix forms of the arithmetic-geometric mean inequality, SIAM J. Matrix Anal. Appl., 14 (1993) 132-136.
35. R. Bhatia and C. Davis, A better bound on the variance, Am. Math. Monthly, 107 (2000) 602-608.
36. R. Bhatia and C. Davis, More operator versions of the Schwarz inequality, Commun. Math. Phys., 215 (2000) 239-244.
37. R. Bhatia and J. A. Dias da Silva, Variation of induced linear operators, Linear Algebra Appl., 341 (2002) 391-402.
38. R. Bhatia and D. Drissi, Generalised Lyapunov equations and positive definite functions, SIAM J. Matrix Anal. Appl., 27 (2005) 103-114.
39. R. Bhatia and L. Elsner, Positive linear maps and the Lyapunov equation, Oper. Theory: Adv. Appl., 130 (2001) 107-120.
40. R. Bhatia and S. Friedland, Variation of Grassman powers and spectra, Linear Algebra Appl., 40 (1981) 1-18.
41. R. Bhatia and J.A.R. Holbrook, Fréchet derivatives of the power function, Indiana Univ. Math. J., 49 (2000) 1155-1173.
42. R. Bhatia and J.A.R. Holbrook, Noncommutative geometric means, Math. Intelligencer, 28 (2006) 32-39.
43. R. Bhatia and J.A.R. Holbrook, Riemannian geometry and matrix geometric means, Linear Algebra Appl., 413 (2006) 594-618.
44. R. Bhatia and F. Kittaneh, On the singular values of a product of operators, SIAM J. Matrix Anal. Appl., 11 (1990) 272-277.
45. R. Bhatia and F. Kittaneh, Norm inequalities for positive operators, Lett. Math. Phys. 43 (1998) 225-231.
46. R. Bhatia and F. Kittaneh, Notes on matrix arithmetic-geometric mean inequalities, Linear Algebra Appl., 308 (2000) 203-211.
47. R. Bhatia and H. Kosaki, Mean matrices and infinite divisibility, Linear Algebra Appl., To appear.
48. R. Bhatia and K. R. Parthasarathy, Positive definite functions and operator inequalities, Bull. London Math. Soc., 32 (2000) 214-228.
49. R. Bhatia, D. Singh, and K. B. Sinha, Differentiation of operator functions and perturbation bounds, Commun. Math. Phys., 191 (1998) 603-611.
50. R. Bhatia and K. B. Sinha, Variation of real powers of positive operators, Indiana Univ. Math. J., 43 (1994) 913-925.
51. R. Bhatia and K. B. Sinha, Derivations, derivatives and chain rules, Linear Algebra Appl., 302/303 (1999) 231-244.
52. M. Sh. Birman and M. Z. Solomyak, Double Stieltjes operator integrals (English translation), in Topics in Mathematical Physics Vol. 1, Consultant Bureau, New York, 1967.
53. S. Bochner, Vorlesungen ̈uber Fouriersche Integrale, Akademie-Verlag, Berlin, 1932.
54. L. Breiman, Probability, Addison-Wesley, 1968.
55. M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, 1999.
56. H. J. Carlin and G. A. Noble, Circuit properties of coupled dispersive lines with applications to wave guide modelling, in Proc. Network and Signal Theory, J. K. Skwirzynski and J. O. Scanlan, eds., Peter Pergrinus, 1973, pp. 258-269.
57. E. Cartan, Groupes simples clos et ouverts et géometrie Riemannienne, J. Math. Pures Appl., 8 (1929) 1-33.
58. M.-D. Choi, Positive linear maps on C*-algebras, Canadian J. Math., 24 (1972) 520-529.
59. M.-D. Choi, A Schwarz inequality for positive linear maps on C*-algebras, Illinois J. Math., 18 (1974) 565-574.
60. M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl., 10 (1975) 285-290.
61. M.-D. Choi, Some assorted inequalities for positive linear maps on C*-algebras, J. Oper. Theory, 4 (1980) 271-285.
62. M.-D. Choi, Positive linear maps, in Operator Algebras and Applications, Part 2, R. Kadison ed., American Math. Society, 1982.
63. G. Corach, H. Porta, and L. Recht, Geodesics and operator means in the space of positive operators, Int. J. Math., 4 (1993) 193-202.
64. G. Corach, H. Porta and L. Recht, Convexity of the geodesic distance on spaces of positive operators, Illinois J. Math., 38 (1994) 87-94.
65. R. Cuppens, Decomposition of Multivariate Probabilities, Academic Press, 1975.
66. Ju. L. Daleckii and S. G. Krein, Formulas of differentiation according to a parameter of functions of Hermitian operators, Dokl. Akad. Nauk SSSR, 76 (1951) 13-16.
67. K. R. Davidson and J.A.R. Holbrook, Numerical radii of zero-one matrices, Michigan Math. J., 35 (1988) 261-267.
68. C. Davis, A Schwarz inequality for convex operator functions, Proc. Am. Math. Soc., 8 (1957) 42-44.
69. C. Davis, Notions generalizing convexity for functions defined on spaces of matrices, in Proc. Symposia Pure Math., Vol.VII, Convexity, American Math. Soc., 1963.
70. J. De Pillis, Transformations on partitioned matrices, Duke Math. J., 36 (1969) 511-515.
71. D. Drissi, Sharp inequalities for some operator means, SIAM J. Matrix Anal. Appl., To appear.
72. S. W. Drury, S. Liu, C.-Y. Lu, S. Puntanen, and G.P.H. Styan, Some comments on several matrix inequalities with applications to canonical correlations: historical background and recent developments, Sankhy-a, Series A, 64 (2002) 453-507.
73. E. G. Effros and Z.-J. Ruan, Operator Spaces, Oxford University Press, 2000.
74. W. G. Faris, Review of S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, SIAM Rev., 47 (2005) 379-380.
75. M. Fiedler and V. Pták, A new positive definite geometric mean of two positive definite matrices, Linear Algebra Appl., 251 (1997) 1-20.
76. (p+2r)/q for r≥0, p≥0, q≥1 with (1 + 2r)q≥p+ 2r, Proc. Am. Math. Soc. 101 (1987) 85-88.
77. F. Gantmacher, Matrix Theory, Chelsea, 1977.
78. B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, 1954.
79. R. Goldberg, Fourier Transforms, Cambridge University Press, 1961.
80. S. Golden, Lower bounds for the Helmholtz function, Phys. Rev. B, 137 (1965) 1127-1128.
81. G. H. Golub and C. F. Van Loan, Matrix Computations, Third Edition, Johns Hopkins University Press, 1996.
82. U. Haagerup, Decomposition of completely bounded maps on operator algebras, unpublished report.
83. P. R. Halmos, A Hilbert Space Problem Book, Second Edition, Springer, 1982.
84. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Second Edition, Cambridge University Press, 1952.
85. E. Haynsworth, Determination of the inertia of a partitioned Hermitian matrix, Linear Algebra Appl., 1 (1968) 73-81.
86. E. Heinz, Beitr̈age zur Sẗorungstheorie der Spektralzerlegung, Math. Ann., 123 (1951) 415-438.
87. C. S. Herz, Fonctions opérant sur les fonctions définies-positives, Ann. Inst. Fourier (Grenoble), 13 (1963) 161-180.
88. F. Hiai, Log-majorizations and norm inequalities for exponential operators, Banach Center Publications Vol. 38, pp. 119-181.
89. F. Hiai and H. Kosaki, Means for matrices and comparison of their norms, Indiana Univ. Math. J., 48 (1999) 899-936.
90. F. Hiai and H. Kosaki, Comparison of various means for operators, J. Funct. Anal., 163 (1999) 300-323.
91. F. Hiai and H. Kosaki, Means of Hilbert Space Operators, Lecture Notes in Mathematics Vol. 1820, Springer, 2003.
92. F. Hiai and D. Petz, The Golden-Thompson trace inequality is complemented, Linear Algebra Appl., 181 (1993) 153-185.
93. N. J. Higham, Accuracy and Stability of Numerical Algorithms, Second Edition, SIAM, 2002.
94. R. A. Horn, On boundary values of a schlicht mapping, Proc. Am. Math. Soc., 18 (1967) 782-787.
95. R. A. Horn, The Hadamard product, in Matrix Theory and Applications, C. R. Johnson, ed., American Math. Society, 1990.
96. R. A. Horn, Norm bounds for Hadamard products and the arithmeticgeometric mean inequality for unitarily invariant norms, Linear Algebra Appl., 223/224 (1995) 355-361.
97. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985.
98. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.
99. D. K. Jocić, Cauchy-Schwarz and means inequalities for elementary operators into norm ideals, Proc. Am. Math. Soc., 126 (1998) 2705-2711.
100. D. K. Jocić, Cauchy-Schwarz norm inequalities for weak*-integrals of operator valued functions, J. Funct. Anal., 218 (2005) 318-346.
101. C. R. Johnson, Matrix completion problems: a survey, Proc., Symposia in Applied Mathematics Vol. 40, American Math. Society, 1990.
102. C. R. Johnson and C. J. Hillar, Eigenvalues of words in two positive definite letters, SIAM J. Matrix Anal. Appl., 23 (2002) 916-928.
103. Z. J. Jurek and M. Yor, Self decomposable laws associated with hyperbolic functions, Probability and Mathematical Statistics, 24(2004), 181-190.
104. R. Kadison, A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of Math., 56 (1952) 494-503.
105. H. Kosaki, Arithmetic-geometric mean and related inequalities for operators, J. Funct. Anal., 15 (1998) 429-451.
106. H. Kosaki, On infinite divisibility of positive definite functions, preprint, 2006.
107. K. Kraus, General state changes in quantum theory, Ann. Phys., 64 (1971) 311-335.
108. K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Lecture Notes in Physics Vol. 190, Springer, 1983.
109. F. Kubo and T. Ando, Means of positive linear operators, Math. Ann., 246 (1980) 205-224.
110. A. Kurosh, Higher Algebra, Mir Publishers, 1972.
111. M. K. Kwong, On the definiteness of the solutions of certain matrix equations, Linear Algebra Appl., 108 (1988) 177-197.
112. M. K. Kwong, Some results on matrix monotone functions, Linear Algebra Appl., 118 (1989) 129-153.
113. P. Lancaster and L. Rodman, The Algebraic Riccati Equation, Oxford University Press, 1995.
114. P. Lancaster and M. Tismenetsky, The Theory of Matrices, Second Edition, Academic Press, 1985.
115. S. Lang, Fundamentals of Differential Geometry, Springer, 1999.
116. J. D. Lawson and Y. Lim, The geometric mean, matrices, metrics and more, Am. Math. Monthly, 108 (2001) 797-812.
117. P. Lax, Linear Algebra, John Wiley, 1997.
118. P. Lévy, Théorie de l' Addition des Variables Aléatoires, Gauthier-Villars, 1937.
119. E. H. Lieb, Convex trace functions and the Wigner-Yanase-Dyson conjecture, Adv. Math., 11 (1973) 267-288.
120. E. H. Lieb, Inequalities, Selecta of Elliot H. Lieb, M. Loss and M. B. Ruskai eds., Springer, 2002.
121. E. H. Lieb and M. B. Ruskai, A fundamental property of quantum-mechanical entropy, Phys. Rev. Lett., 30 (1973) 434-436.
122. E. H. Lieb and M. B. Ruskai, Proof of the strong subadditivity of quantummechanical entropy, J. Math. Phys., 14 (1973) 1938-1941.
123. E. H. Lieb and M. B. Ruskai, Some operator inequalities of the Schwarz type, Adv. Math., 12 (1974) 269-273.
124. E. H. Lieb and R. Seiringer, Equivalent forms of the Bessis-Moussa-Villani conjecture, J. Stat. Phys., 115 (2004) 185-190.
125. E. H. Lieb and J. Yngvason, A guide to entropy and the second law of thermodynamics, Notices Am. Math. Soc., 45 (1998) 571-581.
126. E. H. Lieb and J. Yngvason, The mathematical structure of the second law of thermodynamics, in Current Developments in Mathematics, 2001, International Press, 2002.
127. G. Lindblad, Entropy, information and quantum measurements, Commun. Math. Phys., 33 (1973) 305-322.
128. G. Lindblad, Expectations and entropy inequalities for finite quantum systems, Commun. Math. Phys., 39 (1974) 111-119.
129. G. Lindblad, Completely positive maps and entropy inequalities, Commun. Math. Phys., 40 (1975) 147-151.
130. M. Loeve, Probability Theory, Van Nostrand, 1963.
131. K. L̈owner, ̈Uber monotone Matrixfunctionen, Math. Z., 38 (1934) 177-216.
132. E. Lukacs, Characteristic Functions, Griffin, 1960.
133. M. Marcus and W. Watkins, Partitioned Hermitian matrices, Duke Math. J., 38 (1971) 237-249.
134. R. Mathias, An arithmetic-geometric mean inequality involving Hadamard products, Linear Algebra Appl., 184 (1993) 71-78.
135. R. Mathias, A note on: "More operator versions of the Schwarz inequality," Positivity, 8 (2004) 85-87.
136. W. H. McAdams, Heat Transmission, Third Edition, McGraw-Hill, 1954.
137. R. Merris, Trace functions I, Duke Math. J., 38 (1971) 527-530.
138. P. A. Meyer, Quantum Probability for Probabilists, Lecture Notes in Mathematics Vol. 1538, Springer, 1993.
139. M. Moakher, A differential geometric approach to the geometric mean of symmetric positive-definite matrices, SIAM J. Matrix Anal. Appl., 26 (2005) 735-747.
140. M. Moakher, On the averaging of symmetric positive-definite tensors, preprint (2005).
141. P. Moussa, On the representation of Tr " e(A-ΛB) " as a Laplace transform, Rev. Math. Phys., 12 (2000) 621-655.
142. V. M̈uller, The numerical radius of a commuting product, Michigan Math. J., 35 (1988) 255-260.
143. J. von Neumann, Thermodynamik quantenmechanischer Gesamtheiten, G̈ottingen Nachr., 1927, pp. 273-291.
144. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
145. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.
146. M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer, 1993.
147. K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkḧauser, 1992.
148. K. R. Parthasarathy and K. Schmidt, Positive Definite Kernels, Continuous Tensor Products and Central Limit Theorems of Probability Theory, Lecture Notes in Mathematics Vol. 272, Springer, 1972.
149. V. Paulsen, Completely Bounded Maps and Dilations, Longman, 1986.
150. V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2002.
151. V. I. Paulsen, S. C. Power, and R. R. Smith, Schur products and matrix completions, J. Funct. Anal., 85 (1989) 151-178.
152. D. Petz and R. Temesi, Means of positive numbers and matrices, SIAM J. Matrix Anal. Appl., 27 (2005) 712-720.
153. G. Pisier, Introduction to Operator Space Theory, Cambridge University Press, 2003.
154. G. Pólya, Remarks on characteristic functions, Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, 1949, pp. 115-123.
155. W. Pusz and S. L. Woronowicz, Functional calculus for sesquilinear forms and the purification map, Rep. Math. Phys., 8 (1975) 159-170.
156. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vols. I, II, Academic Press, 1972, 1975.
157. B. Russo and H. A. Dye, A note on unitary operators in C*-algebras, Duke Math. J., 33 (1966) 413-416.
158. Z. Sasvári, Positive Definite and Definitizable Functions, Akademie-Verlag, Berlin, 1994.
159. I. Schur, Bemerkungen zur Theorie der beschr̈ankten Bilinearformen mit unendlich vielen Ver̈anderlichen, J. Reine Angew. Math., 140 (1911), 1-28.
160. I. Schur, ̈ Uber Potenzreihen die im Innern des Einheitskreises beschr̈ankt sind I], J. Reine Angew. Math., 147 (1917) 205-232.
161. L. Schwartz, Théorie des Distributions, Hermann, 1954.
162. I. Segal, Notes towards the construction of nonlinear relativistic quantum fields III, Bull. Am. Math. Soc., 75 (1969) 1390-1395.
163. C. Shannon, Mathematical theory of communication, Bell Sys. Tech. J., 27 (1948) 379-423.
164. P. W. Shor, Equivalence of additivity questions in quantum information theory, Commun. Math. Phys., 246 (2004) 453-472.
165. B. Simon, Trace Ideals and Their Applications, Second Edition, American Mathematical Society, 2005.
166. J. M. Steele, The Cauchy-Schwarz Master Class, Math. Association of America, 2004.
167. J. Stewart, Positive definite functions and generalizations, an historical survey, Rocky Mountain J. Math., 6 (1976) 409-434.
168. W. F. Stinespring, Positive functions on C*-algebras, Proc. Am. Math. Soc., 6 (1955) 211-216.
169. E. Størmer, Positive linear maps of operator algebras, Acta Math., 110 (1963) 233-278.
170. E. Størmer, Positive linear maps of C*-algebras, in Foundations of Quantum Mechanics and Ordered Linear Spaces, Lecture Notes in Physics Vol. 29, Springer, 1974, pp. 85-106.
171. V. S. Sunder, A noncommutative analogue of DXk≥kXk-1, Linear Algebra Appl., 44 (1982) 87-95.
172. A. Terras, Harmonic Analysis on Symmetric Spaces and Applications II, Springer, 1988.
173. C. J. Thompson, Inequality with applications in statistical mechanics, J. Math. Phys., 6 (1965) 1812-1813.
174. S.-G. Wang and W.-C. Ip, A matrix version of the Wielandt inequality and its applications, Linear Algebra Appl., 296 (1999) 171-181.
175. A. Wehrl, General properties of entropy, Rev. Mod. Phys., 50 (1978) 221-260.
176. A.Wehrl, The many facets of entropy, Rep. Math. Phys., 30 (1991) 119-129.
177. S. L. Woronowicz, Positive maps of low dimensional matrix algebras, Rep. Math. Phys., 10 (1976) 165-183.
178. X. Zhan, Matrix Inequalities, Lecture Notes in Mathematics Vol. 1790, Springer, 2002.
179. F. Zhang, Matrix Theory: Basic Results and Techniques, Springer, 1999.
180. F. Zhang (editor), The Schur Complement and Its Applications, Springer, 2005.