Abstract

A fundamental result of von Neumann’s identifies unitarily invariant matrix norms as symmetric gauge functions of the singular values. Identifying the subdifferential of such a norm is important in matrix approximation algorithms, and in studying the geometry of the corresponding unit ball. We show how to reduce many convex-analytic questions of this kind to questions about the underlying gauge function, via an elegant Fenchel conjugacy formula. This approach also allows such results to be extended to more general unitarily invariant matrix functions.

Author information

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A.S. Lewis

Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1.

Keywords

  • unitarily invariant norm
  • matrix function
  • Fenchel conjugacy
  • subdifferential
  • matrix optimization
  • extreme point
  • exposed point

References

  1. J. Arazy: On the geometry of the unit ball of unitary matrix spaces. Integral Equations and Operator Theory, 4:151–171, 1981.https://doi.org/10.1007/BF01702378Google Scholar
  2. H. Berens and M. Finzel: A continuous selection of the metric projection in matrix spaces. In L. Collatz, G. Meinardus, and G. Nurnberger, editors, Numerical Methods of Approximation Theory, Volume 8. Birkhäser, Basel, 1987.https://doi.org/10.1007/978-3-0348-6656-9_2Google Scholar
  3. E.M. de Sá: Exposed faces and duality for symmetric and unitarily invariant norms. Linear Algebra and its Applications, 197,198:429–450, 1994.https://doi.org/10.1016/0024-3795(94)90499-5Google Scholar
  4. E.M. de Sá: Faces of the unit ball of a unitarily invariant norm. Linear Algebra and its Applications, 197,198:451–493, 1994.https://doi.org/10.1016/0024-3795(94)90500-2Google Scholar
  5. R.A. Horn and C. Johnson: Matrix analysis. Cambridge University Press, Cambridge, U.K., 1985.https://doi.org/10.1017/CBO9780511810817Google Scholar
  6. A.S. Lewis: Convex analysis on the Hermitian matrices. Technical Report CORR 93-33, University of Waterloo, 1993. Accepted by SIAM Journal on Optimization.Google Scholar
  7. C.K. Li and N.K. Tsing: Duality between some linear preserver problems, II. Isometries with respect to c-spectral norms and matrices with fixed singular values. Linear Algebra and its Applications, 110:181–212, 1988.https://doi.org/10.1016/0024-3795(83)90136-2Google Scholar
  8. R.T. Rockafellar: Convex Analysis. Princeton University Press, Princeton, N.J., 1970.Google Scholar
  9. J. von Neumann: Some matrix inequalities and metrization of matric-space. Tomsk University Review, 1:286–300, 1937. In: Collected Works, Pergamon, Oxford, 1962, Volume IV, 205-218.Google Scholar
  10. G.A. Watson: Characterization of the subdifferential of some matrix norms. Linear Algebra and its Applications, 170:33–45, 1992.https://doi.org/10.1016/0024-3795(92)90407-2Google Scholar
  11. G.A. Watson: On matrix approximation problems with Ky Fan k norms. Numerical Algorithms, 5:263–272, 1993.https://doi.org/10.1007/BF02210386Google Scholar
  12. K. Ziętak: On the characterization of the extremal points of the unit sphere of matrices. Linear Algebra and its Applications, 106:57–75, 1988.https://doi.org/10.1016/0024-3795(88)90023-7Google Scholar
  13. K. Ziętak: Subdifferentials, faces and dual matrices. Linear Algebra and its Applications, 185:125–141, 1993.https://doi.org/10.1016/0024-3795(93)90209-7Google Scholar

Suggested citation

A. S. Lewis. “The Convex Analysis of Unitarily Invariant Matrix Functions.” Journal of Convex Analysis 2 (1995), No. 1&2, 173–183. https://doi.org/10.68381/jca02012

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Published by Heldermann Verlag, 1995. Rights now held by Banach Press LLC.