Off-diagonal submatrices of a Hermitian matrix
HTML articles powered by AMS MathViewer
- by Chi-Kwong Li and Yiu-Tung Poon PDF
- Proc. Amer. Math. Soc. 132 (2004), 2849-2856 Request permission
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 $S$.References
- T. Ando, Bloomfield-Watson-Knott type inequalities for eigenvalues, Taiwanese J. Math. 5 (2001), no. 3, 443–469. MR 1849770, DOI 10.11650/twjm/1500574942
- 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
- Anders Skovsted Buch, The saturation conjecture (after A. Knutson and T. Tao), Enseign. Math. (2) 46 (2000), no. 1-2, 43–60. With an appendix by William Fulton. MR 1769536
- Muneo Ch\B{o} and Makoto Takaguchi, Some classes of commuting $n$-tuples of operators, Studia Math. 80 (1984), no. 3, 245–259. MR 783993, DOI 10.4064/sm-80-3-245-259
- Man-Duen Choi, Chi-Kwong Li, and Yiu-Tung Poon, Some convexity features associated with unitary orbits, Canad. J. Math. 55 (2003), no. 1, 91–111. MR 1952327, DOI 10.4153/CJM-2003-004-x
- Jane Day, Wasin So, and Robert C. Thompson, The spectrum of a Hermitian matrix sum, Linear Algebra Appl. 280 (1998), no. 2-3, 289–332. MR 1644987, DOI 10.1016/S0024-3795(98)10019-8
- 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”.
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
- William Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 209–249. MR 1754641, DOI 10.1090/S0273-0979-00-00865-X
- William Fulton, Eigenvalues of majorized Hermitian matrices and Littlewood-Richardson coefficients, Linear Algebra Appl. 319 (2000), no. 1-3, 23–36. Special Issue: Workshop on Geometric and Combinatorial Methods in the Hermitian Sum Spectral Problem (Coimbra, 1999). MR 1799622, DOI 10.1016/S0024-3795(00)00218-4
- Uwe Helmke and Joachim Rosenthal, Eigenvalue inequalities and Schubert calculus, Math. Nachr. 171 (1995), 207–225. MR 1316359, DOI 10.1002/mana.19951710113
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- Alfred Horn, Eigenvalues of sums of Hermitian matrices, Pacific J. Math. 12 (1962), 225–241. MR 140521, DOI 10.2140/pjm.1962.12.225
- Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183, DOI 10.1017/CBO9780511810817
- Roger A. Horn and Charles R. Johnson, Topics in matrix analysis, Cambridge University Press, Cambridge, 1991. MR 1091716, DOI 10.1017/CBO9780511840371
- Alexander A. Klyachko, Stable bundles, representation theory and Hermitian operators, Selecta Math. (N.S.) 4 (1998), no. 3, 419–445. MR 1654578, DOI 10.1007/s000290050037
- Allen Knutson and Terence Tao, The honeycomb model of $\textrm {GL}_n(\textbf {C})$ tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), no. 4, 1055–1090. MR 1671451, DOI 10.1090/S0894-0347-99-00299-4
- Allen Knutson and Terence Tao, Honeycombs and sums of Hermitian matrices, Notices Amer. Math. Soc. 48 (2001), no. 2, 175–186. MR 1811121
- A. Knutson, T. Tao, and C. Woodward, Honeycombs II: Facets of the Littlewood–Richardson cone, to appear.
- Chi-Kwong Li and Roy Mathias, Inequalities on the singular values of an off-diagonal block of a Hermitian matrix, J. Inequal. Appl. 3 (1999), no. 2, 137–142. MR 1733108, DOI 10.1155/S1025583499000090
- Robert C. Thompson and Linda J. Freede, Eigenvalues of partitioned hermitian matrices, Bull. Austral. Math. Soc. 3 (1970), 23–37. MR 265376, DOI 10.1017/S0004972700045615
- A. Zelevinsky, Littlewood–Richardson semigroups, MSRI preprint 1997-044.
Additional Information
- Chi-Kwong Li
- Affiliation: Department of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, Virginia 23187-8795
- MR Author ID: 214513
- Email: ckli@math.wm.edu
- Yiu-Tung Poon
- Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
- MR Author ID: 141040
- Email: ytpoon@iastate.edu
- 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
- © Copyright 2004 American Mathematical Society
- 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
- MathSciNet review: 2063102