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Off-diagonal submatrices of a Hermitian matrix


Authors: Chi-Kwong Li and Yiu-Tung Poon
Journal: Proc. Amer. Math. Soc. 132 (2004), 2849-2856
MSC (2000): Primary 15A18, 15A42
DOI: https://doi.org/10.1090/S0002-9939-04-07072-8
Published electronically: June 2, 2004
MathSciNet review: 2063102
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Abstract: A necessary and sufficient condition is given to a $p\times q$ complex matrix $X$ to be an off-diagonal block of an $n\times n$ Hermitian matrix $C$ with prescribed eigenvalues (in terms of the eigenvalues of $C$ and singular values of $X$). The proof depends on some recent breakthroughs in the study of spectral inequalities on the sum of Hermitian matrices by Klyachko and Fulton. Some interesting geometrical properties of the set $S$ of all such matrices are derived from the main result. These results improve earlier ones that only give partial information for the set $S$.


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  • 1. T. Ando, Bloomfield-Watson-Knott type inequalities for eigenvalues, Taiwanese J. Math. 5 (2001), 443-469. MR 2002f:15022
  • 2. R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1996. MR 98i:15003
  • 3. A.S. Buch, The saturation conjecture (after A. Knutson and T. Tao). With an appendix by William Fulton. Enseign. Math. (2) 46 (2000), no. 1-2, 43-60. MR 2001g:05105
  • 4. M. Cho and M. Takaguchi, Some classes of commuting $m$-tuples of operators, Studia Math. 80 (1984), 245-259. MR 86g:47059
  • 5. M.D. Choi, C.K. Li and Y.T. Poon, Some convexity features associated with unitary orbits, Canad. J. Math. 55 (2003), 91-111. MR 2003m:15051
  • 6. J. Day, W. So and R.C. Thompson, The spectrum of a Hermitian matrix sum, Linear Algebra Appl. 280 (1998), 289-332. MR 99f:15009
  • 7. S.W. Drury, S. Liu, C.-Y. Lu, S. Puntanen, and G.P.H. Styan, Some comments on several matrix inequalities with applications to canonical correlations, to appear in the Special Issue of Sankhya associated with ``An International Conference in Honor of Professor C.R. Rao on the occasion of his 80th Birthday, Statistics: Reflections on the Past and Visions for the Future, The University of Texas at San Antonio, March 2000''.
  • 8. K. Fan and G. Pall, Imbedding conditions for Hermitian and normal matrices, Canad. J. Math. 9 (1957), 298-304. MR 19:6e
  • 9. W. Fulton, Young Tableaux, Cambridge University Press, Cambridge, 1997. MR 99f:05119
  • 10. W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Amer. Math. Soc. (N.S.) 37 (2000), 209-249. MR 2001g:15023
  • 11. W. Fulton, Eigenvalues of majorized Hermitian matrices, and Littlewood-Richardson coefficients, Linear Algebra Appl. 319 (2000), 23-36. MR 2002a:15024
  • 12. U. Helmke and J. Rosenthal, Eigenvalue inequalities and Schubert calculus, Math. Nachr. 171 (1995), 207-225. MR 96b:15039
  • 13. A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math. 76 (1954), 620-630. MR 16:105c
  • 14. A. Horn, Eigenvalues of sums of Hermitian matrices, Pacific J. Math. 12 (1962), 225-241. MR 25:3941
  • 15. R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985. MR 87e:15001
  • 16. R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991. MR 92e:15003
  • 17. A.A. Klyachko, Stable bundles, representation theory and Hermitian operators, Selecta Math. (N.S.) 4 (1998), 419-445. MR 2000b:14054
  • 18. A. Knutson and T. Tao, The honeycomb model of $\operatorname{GL}\sb n(c)$ tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), no. 4, 1055-1090. MR 2000c:20066
  • 19. A. Knutson and T. Tao, Honeycombs and sums of Hermitian matrices, Notices of the Amer. Math. Soc. 48 (2001), no. 2, 175-186. MR 2002g:15020
  • 20. A. Knutson, T. Tao, and C. Woodward, Honeycombs II: Facets of the Littlewood-Richardson cone, to appear.
  • 21. C.K. Li and R. Mathias, Inequalities on the singular values of an off-diagonal block of a Hermitian matrix, J. of Inequalities and Applications 3 (1999), 137-142. MR 2001b:15025
  • 22. R.C. Thompson and L.J. Freede, Eigenvalues of partitioned Hermitian matrices, Bulletin Austral. Math. Soc. 3 (1970), 23-37. MR 42:286
  • 23. A. Zelevinsky, Littlewood-Richardson semigroups, MSRI preprint 1997-044.

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Additional Information

Chi-Kwong Li
Affiliation: Department of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, Virginia 23187-8795
Email: ckli@math.wm.edu

Yiu-Tung Poon
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: ytpoon@iastate.edu

DOI: https://doi.org/10.1090/S0002-9939-04-07072-8
Keywords: Hermitian matrices, singular values, eigenvalues, Littlewood-Richardson rules
Received by editor(s): February 18, 2002
Received by editor(s) in revised form: October 24, 2002
Published electronically: June 2, 2004
Additional Notes: The first author’s research was partially supported by an NSF grant
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2004 American Mathematical Society

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