Extension of Wang-Gong monotonicity result in semisimple Lie groups

• Autorzy: Zachary Sarver , Tin-Yau Tam
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Wang-Gong inequality; positive definite matrices; semisimple Lie groups; log majorization; Kostant’s pre-order

• Wydawca: De Gruyter Open
• Czasopismo: Special Matrices
• Rocznik: 2015
• Tom: 3
• Numer: 1

Daty

otrzymano

2015-09-04

zaakceptowano

2015-10-12

online

2015-10-16

Twórcy

autor

Zachary Sarver

• Department of Mathematics and Statistics, Auburn University,
  Auburn, AL 36849, USA

autor

Tin-Yau Tam

• Department of Mathematics and Statistics, Auburn University,
  Auburn, AL 36849, USA

Bibliografia

• [1] H. Araki, On an inequality of Lieb and Thirring, Lett. Math. Phys. 19, (1990) 167–170.
• [2] K. M. R. Audenaert, On the Araki-Lieb-Thirring inequality, Int. J. Inform. Sys. Sci. 4, (2008) 78–83.
• [3] K. M. R. Audenaert, A Lieb-Thirring inequality for singular values, Linear Algebra Appl. 430, (2009) 3053–3057.
• [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.
• [6] K.J. Le Couteur, Representation of the function tr (exp(A − ʎB)) as a Laplace transform with positive weight and somematrixinequalities, J. Phys. A 13, (1980), 3147–3159.
• [7] S. Helgason, “Differential Geometry, Lie Groups, and Symmetric Spaces”, Academic Press, Inc. [Harcourt Brace Jovanovich,Publishers], New York-London, 1978.
• [8] R. Horn, and C. Johnson, “Matrix analysis (2nd ed.)”, Cambridge University Press, Cambridge, 2013.
• [9] A. W. Knapp, “Lie Groups beyond an Introduction", 2nd ed., Birkhäuser Boston, Inc., Boston, MA, 2002.
• [10] B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. (4) 6 (1973), 413–455.
• [11] E. Lieb, andW. Thirring, in “Studies inMathematical Physics" (Eds. E. Lieb, B. Simon and A.Wightman), p.301–302, PrincetonPress, 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.
• [13] X. Liu, and T.Y. Tam, Extensions of three matrix inequalities to semisimple Lie groups, Special Matrices, 2 (2014), 148–154.
• [14] M. Marcus, An eigenvalues inequality for the product of normal matrices, Amer. Math. Monthly, 63 (1956), 173–174.
• [15] A. W. Marshall, I. Olkin, and B. C. Arnold, “Inequalities: Theory of Majorization and its Applications (2nd ed.)”, Springer,New York, 2011.
• [16] T.Y. Tam, and H. Huang, An extension of Yamamoto’s theorem on the eigenvalues and singular values of a matrix, Journal ofMath. Soc. Japan, 58 (2006), 1197–1202.
• [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.
• [18] B.Y. Wang, and M.P. Gong, Some eigenvalue inequalities for positive semidefinite matrix power products, Linear AlgebraAppl. 184 (1993), 249–260.
• [19] B.Y. Wang, and F. Zhang, Trace and eigenvalue inequalities for ordinary and hadamard products of positive semidefinitehermitian matrices, SIAM J. Matrix Anal. Appl., 16 (1995), 1173–1183.
• [20] X. Zhan, “Matrix Inequalities”, Lecture Notes in Mathematics 1790, Springer-Verlag, Berlin, 2002.

Identyfikatory

DOI

10.1515/spma-2015-0024