On matrix rearrangement inequalities
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- by Rima Alaifari, Xiuyuan Cheng, Lillian B. Pierce and Stefan Steinerberger PDF
- Proc. Amer. Math. Soc. 148 (2020), 1835-1848 Request permission
Abstract:
Given two symmetric and positive semidefinite square matrices $A, B$, is it true that any matrix given as the product of $m$ copies of $A$ and $n$ copies of $B$ in a particular sequence must be dominated in the spectral norm by the ordered matrix product $A^m B^n$? For example, is \begin{equation*} \| AABAABABB \| \leq \| AAAAABBBB \| ? \end{equation*} Drury [Electron J. Linear Algebra 18 (2009), pp. 13–20] has characterized precisely which disordered words have the property that an inequality of this type holds for all matrices $A,B$. However, the $1$-parameter family of counterexamples Drury constructs for these characterizations is comprised of $3 \times 3$ matrices, and thus as stated the characterization applies only for $N \times N$ matrices with $N \geq 3$. In contrast, we prove that for $2 \times 2$ matrices, the general rearrangement inequality holds for all disordered words. We also show that for larger $N \times N$ matrices, the general rearrangement inequality holds for all disordered words for most $A,B$ (in a sense of full measure) that are sufficiently small perturbations of the identity.References
- W. Albar, M. Junge, and M. Zhao, On the symmetrized arithmetic-geometric mean inequality for operators, arXiv:1803.02435
- Tsuyoshi Ando, Fumio Hiai, and Kazuyoshi Okubo, Trace inequalities for multiple products of two matrices, Math. Inequal. Appl. 3 (2000), no. 3, 307–318. MR 1768492, DOI 10.7153/mia-03-32
- E. Andruchow, G. Corach, and D. Stojanoff, Geometrical significance of Löwner-Heinz inequality, Proc. Amer. Math. Soc. 128 (2000), no. 4, 1031–1037. MR 1636922, DOI 10.1090/S0002-9939-99-05085-6
- Rajendra Bhatia, Matrix analysis, Graduate Texts in Mathematics, vol. 169, Springer-Verlag, New York, 1997. MR 1477662, DOI 10.1007/978-1-4612-0653-8
- Rajendra Bhatia, Positive definite matrices, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2007. MR 2284176
- G. Corach, H. Porta, and L. Recht, An operator inequality, Linear Algebra Appl. 142 (1990), 153–158. MR 1077981, DOI 10.1016/0024-3795(90)90263-C
- H. O. Cordes, A matrix inequality, Proc. Amer. Math. Soc. 11 (1960), 206–210. MR 112048, DOI 10.1090/S0002-9939-1960-0112048-X
- H. O. Cordes, Spectral theory of linear differential operators and comparison algebras, London Mathematical Society Lecture Note Series, vol. 76, Cambridge University Press, Cambridge, 1987. MR 890743, DOI 10.1017/CBO9780511662836
- J. Dixmier, Sur une inégalité de E. Heinz, Math. Ann. 126 (1953), 75–78 (French). MR 56200, DOI 10.1007/BF01343151
- S. W. Drury, Operator norms of words formed from positive-definite matrices, Electron. J. Linear Algebra 18 (2009), 13–20. MR 2471192, DOI 10.13001/1081-3810.1290
- Masatoshi Fujii and Takayuki Furuta, Löwner-Heinz, Cordes and Heinz-Kato inequalities, Math. Japon. 38 (1993), no. 1, 73–78. MR 1204185
- Takayuki Furuta, Norm inequalities equivalent to Löwner-Heinz theorem, Rev. Math. Phys. 1 (1989), no. 1, 135–137. MR 1041534, DOI 10.1142/S0129055X89000079
- Takayuki Furuta, Invitation to linear operators, Taylor & Francis Group, London, 2001. From matrices to bounded linear operators on a Hilbert space. MR 1978629, DOI 10.1201/b16820
- Fritz Gesztesy, Yuri Latushkin, Fedor Sukochev, and Yuri Tomilov, Some operator bounds employing complex interpolation revisited, Operator semigroups meet complex analysis, harmonic analysis and mathematical physics, Oper. Theory Adv. Appl., vol. 250, Birkhäuser/Springer, Cham, 2015, pp. 213–239. MR 3468218, DOI 10.1007/978-3-319-18494-4_{1}4
- G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
- Erhard Heinz, Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann. 123 (1951), 415–438 (German). MR 44747, DOI 10.1007/BF02054965
- Erhard Heinz, On an inequality for linear operators in a Hilbert space, Report of an international conference on operator theory and group representations, Arden House, Harriman, N. Y., 1955, National Academy of Sciences-National Research Council, Washington, D.C., 1955, pp. 27–29. Publ. 387. MR 0079139
- Arie Israel, Felix Krahmer, and Rachel Ward, An arithmetic-geometric mean inequality for products of three matrices, Linear Algebra Appl. 488 (2016), 1–12. MR 3419770, DOI 10.1016/j.laa.2015.09.013
- Tosio Kato, Notes on some inequalities for linear operators, Math. Ann. 125 (1952), 208–212. MR 53390, DOI 10.1007/BF01343117
- Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225, DOI 10.1090/gsm/014
- Karl Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), no. 1, 177–216 (German). MR 1545446, DOI 10.1007/BF01170633
- Albert W. Marshall, Ingram Olkin, and Barry C. Arnold, Inequalities: theory of majorization and its applications, 2nd ed., Springer Series in Statistics, Springer, New York, 2011. MR 2759813, DOI 10.1007/978-0-387-68276-1
- A. McIntosh, Heinz inequalities and perturbation of spectral families, Macquarie Mathematics Reports, Macquarie University, 1979.
- Lucijan Plevnik, On a matrix trace inequality due to Ando, Hiai and Okubo, Indian J. Pure Appl. Math. 47 (2016), no. 3, 491–500. MR 3552962, DOI 10.1007/s13226-016-0180-9
- B. Recht and C. Ré, Beneath the Valley of the Noncommutative Arithmetic-Geometric Mean Inequality: Conjectures, Case Studies, and Consequences, Proceedings of COLT. 2012
- Barry Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. MR 2154153, DOI 10.1090/surv/120
- Stefan Steinerberger, Refined Heinz-Kato-Löwner inequalities, J. Spectr. Theory 9 (2019), no. 1, 1–20. MR 3900778, DOI 10.4171/JST/239
- Xingzhi Zhan, Matrix inequalities, Lecture Notes in Mathematics, vol. 1790, Springer-Verlag, Berlin, 2002. MR 1927396, DOI 10.1007/b83956
- Teng Zhang, A note on the matrix arithmetic-geometric mean inequality, Electron. J. Linear Algebra 34 (2018), 283–287. MR 3841395, DOI 10.13001/1081-3810.3555
Additional Information
- Rima Alaifari
- Affiliation: Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
- MR Author ID: 1019553
- Email: rima.alaifari@math.ethz.ch
- Xiuyuan Cheng
- Affiliation: Department of Mathematics, Duke University, 120 Science Drive, Durham, North Carolina 27708
- MR Author ID: 892484
- Email: xiuyuan.cheng@duke.edu
- Lillian B. Pierce
- Affiliation: Department of Mathematics, Duke University, 120 Science Drive, Durham, North Carolina 27708
- MR Author ID: 757898
- Email: pierce@math.duke.edu
- Stefan Steinerberger
- Affiliation: Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, Connecticut 06511
- MR Author ID: 869041
- ORCID: 0000-0002-7745-4217
- Email: stefan.steinerberger@yale.edu
- Received by editor(s): April 17, 2019
- Received by editor(s) in revised form: August 22, 2019, and August 25, 2019
- Published electronically: February 13, 2020
- Additional Notes: The second author was partially supported by the NSF (DMS-1818945, DMS-1820827).
The third author was partially supported by CAREER grant NSF DMS-1652173 and the Alfred P. Sloan Foundation.
The fourth author was partially supported by the NSF (DMS-1763179) and the Alfred P. Sloan Foundation. - Communicated by: Stephan Ramon Garcia
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 1835-1848
- MSC (2010): Primary 15A45, 47A30, 47A63; Secondary 39B42
- DOI: https://doi.org/10.1090/proc/14831
- MathSciNet review: 4078071