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Keywords:
geometric mean; positive definite matrix; log majorization; geodesics; geodesically convex; geodesic convex hull; unitarily invariant norm
geometric mean; positive definite matrix; log majorization; geodesics; geodesically convex; geodesic convex hull; unitarily invariant norm
Summary:
We study some geometric properties associated with the $t$-geometric means $A\sharp _{t}B := A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2}$ of two $n\times n$ positive definite matrices $A$ and $B$. Some geodesical convexity results with respect to the Riemannian structure of the $n\times n$ positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding $m$ pairs of positive definite matrices is posted.
References:
[1] Ando, T.: Concavity of certain maps on positive definite matrices and applications to Hadamard products. Linear Algebra Appl. 26 (1979), 203-241. DOI 10.1016/0024-3795(79)90179-4 | MR 0535686 | Zbl 0495.15018
[2] Ando, T., Hiai, F.: Log majorization and complementary Golden-Thompson type inequalities. Linear Algebra Appl. 197/198 (1994), 113-131. MR 1275611 | Zbl 0793.15011
[3] Araki, H.: On an inequality of Lieb and Thirring. Lett. Math. Phys. 19 (1990), 167-170. DOI 10.1007/BF01045887 | MR 1039525 | Zbl 0705.47020
[4] Audenaert, K. M. R.: A norm inequality for pairs of commuting positive semidefinite matrices. Electron. J. Linear Algebra (electronic only) 30 (2015), 80-84. MR 3318430 | Zbl 1326.15030
[5] Bhatia, R.: The Riemannian mean of positive matrices. Matrix Information Geometry Springer, Berlin F. Nielsen et al. (2013), 35-51. DOI 10.1007/978-3-642-30232-9_2 | MR 2964446 | Zbl 1271.15019
[6] Bhatia, R.: Postitive Definite Matrices. Princeton Series in Applied Mathematics Princeton University Press, Princeton (2007). MR 3443454
[7] Bhatia, R.: Matrix Analysis. Graduate Texts in Mathematics 169 Springer, New York (1997). MR 1477662
[8] Bhatia, R., Grover, P.: Norm inequalities related to the matrix geometric mean. Linear Algebra Appl. 437 (2012), 726-733. MR 2921731 | Zbl 1252.15023
[9] Bourin, J.-C.: Matrix subadditivity inequalities and block-matrices. Int. J. Math. 20 (2009), 679-691. DOI 10.1142/S0129167X09005509 | MR 2541930 | Zbl 1181.15030
[10] Bourin, J.-C., Uchiyama, M.: A matrix subadditivity inequality for {$f(A+B)$} and {$f(A)+f(B)$}. Linear Algebra Appl. 423 (2007), 512-518. MR 2312422 | Zbl 1123.15013
[11] Fiedler, M., Pták, V.: A new positive definite geometric mean of two positive definite matrices. Linear Algebra Appl. 251 (1997), 1-20. MR 1421263 | Zbl 0872.15014
[12] Hayajneh, S., Kittaneh, F.: Trace inequalities and a question of Bourin. Bull. Aust. Math. Soc. 88 (2013), 384-389. DOI 10.1017/S0004972712001104 | MR 3189289 | Zbl 1287.47011
[13] Lin, M.: Remarks on two recent results of Audenaert. Linear Algebra Appl. 489 (2016), 24-29. MR 3421835 | Zbl 1326.15033
[14] Lin, M.: Inequalities related to {$2\times2$} block PPT matrices. Oper. Matrices 9 (2015), 917-924. DOI 10.7153/oam-09-54 | MR 3447594
[15] Marshall, A. W., Olkin, I., Arnold, B. C.: Inequalities: Theory of Majorization and Its Applications. Springer Series in Statistics Springer, New York (2011). MR 2759813 | Zbl 1219.26003
[16] Papadopoulos, A.: Metric Spaces, Convexity and Nonpositive Curvature. IRMA Lectures in Mathematics and Theoretical Physics 6 European Mathematical Society, Zürich (2005). MR 2132506 | Zbl 1115.53002
[17] Pusz, W., Woronowicz, S. L.: Functional calculus for sesquilinear forms and the purification map. Rep. Math. Phys. 8 (1975), 159-170. DOI 10.1016/0034-4877(75)90061-0 | MR 0420302 | Zbl 0327.46032
[18] Thompson, R. C.: Singular values, diagonal elements, and convexity. SIAM J. Appl. Math. 32 (1977), 39-63. DOI 10.1137/0132003 | MR 0424847 | Zbl 0361.15009
[19] Zhan, X.: Matrix Inequalities. Lecture Notes in Mathematics 1790 Springer, Berlin (2002). DOI 10.1007/b83956 | MR 1927396 | Zbl 1018.15016
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