We extend a monotonicity result of Wang and Gong on the product of positive definite matrices in the context of semisimple Lie groups. A similar result on singular values is also obtained.
@article{bwmeta1.element.doi-10_1515_spma-2015-0024,
author = {Zachary Sarver and Tin-Yau Tam},
title = {Extension of Wang-Gong monotonicity result in semisimple Lie groups},
journal = {Special Matrices},
volume = {3},
year = {2015},
zbl = {1327.15043},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_spma-2015-0024}
}
Zachary Sarver; Tin-Yau Tam. Extension of Wang-Gong monotonicity result in semisimple Lie groups. Special Matrices, Tome 3 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_spma-2015-0024/
[1] H. Araki, On an inequality of Lieb and Thirring, Lett. Math. Phys. 19, (1990) 167–170. | Zbl 0705.47020
[2] K. M. R. Audenaert, On the Araki-Lieb-Thirring inequality, Int. J. Inform. Sys. Sci. 4, (2008) 78–83. | Zbl 1156.15008
[3] K. M. R. Audenaert, A Lieb-Thirring inequality for singular values, Linear Algebra Appl. 430, (2009) 3053–3057. | Zbl 1168.15014
[4] R. Bhatia, “Matrix Analysis", Springer-Verlag, New York, 1997.
[5] P.J. Bushell, and G.B. Trustrum, Trace inequalities for positive definite matrix power products, Linear Algebra Appl. 132 (1990), 173–178. | Zbl 0701.15015
[6] K.J. Le Couteur, Representation of the function tr (exp(A − ʎB)) as a Laplace transform with positive weight and somematrix inequalities, J. Phys. A 13, (1980), 3147–3159. | Zbl 0446.44002
[7] S. Helgason, “Differential Geometry, Lie Groups, and Symmetric Spaces”, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. | Zbl 0451.53038
[8] R. Horn, and C. Johnson, “Matrix analysis (2nd ed.)”, Cambridge University Press, Cambridge, 2013. | Zbl 1267.15001
[9] A. W. Knapp, “Lie Groups beyond an Introduction", 2nd ed., Birkhäuser Boston, Inc., Boston, MA, 2002. | Zbl 1075.22501
[10] B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. (4) 6 (1973), 413–455. | Zbl 0293.22019
[11] E. Lieb, andW. Thirring, in “Studies inMathematical Physics" (Eds. E. Lieb, B. Simon and A.Wightman), p.301–302, Princeton Press, 1976.
[12] X. Liu, M. Liao, and T.Y. Tam, Geometric mean for symmetric spaces of noncompact type, Journal of Lie Theory 24 (2014), 725–736. | Zbl 1331.15011
[13] X. Liu, and T.Y. Tam, Extensions of three matrix inequalities to semisimple Lie groups, Special Matrices, 2 (2014), 148–154. | Zbl 1308.15019
[14] M. Marcus, An eigenvalues inequality for the product of normal matrices, Amer. Math. Monthly, 63 (1956), 173–174. | Zbl 0070.01304
[15] A. W. Marshall, I. Olkin, and B. C. Arnold, “Inequalities: Theory of Majorization and its Applications (2nd ed.)”, Springer, New York, 2011. | Zbl 1219.26003
[16] T.Y. Tam, and H. Huang, An extension of Yamamoto’s theorem on the eigenvalues and singular values of a matrix, Journal of Math. Soc. Japan, 58 (2006), 1197–1202. | Zbl 1117.15008
[17] T.Y. Tam, Some exponential inequalities for semisimple Lie groups, A chapter of “Operators, Matrices and Analytic Functions”, 539–552, Oper. Theory Adv. Appl. 202, Birkhäuser Verlag, Basel, 2010. | Zbl 1192.15008
[18] B.Y. Wang, and M.P. Gong, Some eigenvalue inequalities for positive semidefinite matrix power products, Linear Algebra Appl. 184 (1993), 249–260. | Zbl 0773.15009
[19] B.Y. Wang, and F. Zhang, Trace and eigenvalue inequalities for ordinary and hadamard products of positive semidefinite hermitian matrices, SIAM J. Matrix Anal. Appl., 16 (1995), 1173–1183. | Zbl 0855.15009
[20] X. Zhan, “Matrix Inequalities”, Lecture Notes in Mathematics 1790, Springer-Verlag, Berlin, 2002. | Zbl 1018.15016