|
Generalized numerical ranges, joint positive definiteness and multiple eigenvalues
Author(s):
Yiu
Tung
Poon
Journal:
Proc. Amer. Math. Soc.
125
(1997),
1625-1634.
MSC (1991):
Primary 15A60;
Secondary 47A12
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
We prove a convexity theorem on a generalized numerical range that combines and generalizes the following results: 1) Friedland and Loewy's result on the existence of a nonzero matrix with multiple first eigenvalue in subspaces of hermitian matrices, 2) Bohnenblust's result on joint positive definiteness of hermitian matrices, 3) the Toeplitz-Hausdorff Theorem on the convexity of the classical numerical range and its various generalizations by Au-Yeung, Berger, Brickman, Halmos, Poon, Tsing and Westwick.
References:
- [ALP]
- J. F. Adams, P. D. Lax and R. S. Phillips, On matrices whose real linear combinations are nonsingular, Proc. Amer. Math. Soc. 16 (1965), 318-322. MR 31:3432
- [A1]
- Y. H. Au-Yeung, A simple proof of the convexity of the field of values defined by two hermitian forms, Aequations Math. 12 (1975), 82-83. MR 51:3200
- [A2]
- -, Some results on generalized numerical ranges and related topics, Current trends in matrix theory (F. Uhlig and R. Grone, eds.), North-Holland, New York, 1987, pp. (23-28). MR 88f:15041
- [AP]
- Y. H. Au-Yeung and Y. T. Poon, A remark on the convexity and positive definiteness concerning hermitian matrices, Southeast Asian Bull. Math. 3 (1979), 85-92. MR 81c:15026
- [AT1]
- Y. H. Au-Yeung and N. K. Tsing, Some theorems on the generalized numerical ranges, Linear and Multilinear Algebra 15 (1984), 3-11. MR 85c:15039
- [AT2]
- -, An extension of the Hausdorff-Toeplitz theorem on the numerical range, Proc. Amer. Math. Soc. 89 (1983), 215-218. MR 85f:15021
- [Be]
- C. A. Berger, Normal dilations, Ph. D. Thesis, Cornell Univ., 1963.
- [Bo]
- F. Bohnenblust, Joint positiveness of matrices, unpublished manuscript.
- [Br]
- L. Brickman, On the field of values of a matrix, Proc. Amer. Math. Soc. 12 (1961), 61-66. MR 23:A160
- [D]
- Doyle, Analysis of feedback systems with structured uncertainties, Proc. IEEE-D 129 (1982), 242-250. MR 84a:93035
- [FT]
- M. K. H. Fan and A. L. Tits, m-form numerical range and the computation of the structured singular value, IEEE Trans. Automat. Control 33 (1988), 284-289. MR 89b:93051
- [FL]
- S. Friedland and R. Loewy, Subspaces of symmetric matrices containing matrices with a multiple first eigenvalue, Pacific J. Math. 62 (1976), 389-399. MR 54:2698
- [Hal1]
- P. Halmos, Numerical ranges and normal dialtions, Acta Sci. Math. (Szeged) 25 (1964), 1-5. MR 30:1399
- [Hal2]
- -, A Hilbert space problem book, Von Nostrand, New York, 1967. MR 34:8178
- [HLP]
- G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge University Press, London, 1934.
- [Hau]
- F. Hausdorff, Der Wertvorrat einer Bilinearform, Math Z. 3 (1919), 314-316.
- [Ho]
- A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math. 76 (1954), 620-630. MR 16:105c
- [HJ]
- R. A. Horn and C. A. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985. MR 87e:15001
- [Le]
- H. C. Lee, Eigenvalues and canonical forms of matrices with quaternion coefficients, Proc. Roy. Irish. Acad. Sect. A 52 (1949), 253-260. MR 12:153i
- [Li]
- C. K. Li, Matrices with some extremal properties, Linear Algebra Appl. 101 (1988), 255-267. MR 89c:15025
- [MO]
- A. W. Marshall and I. Olkin, Inequalities: Theory of majorization and its applications, Academic Press, New York, 1979. MR 81b:00002
- [P1]
- Y. T. Poon, Another proof of a result of Westwick, Linear and Multilinear Algebra 9 (1980), 35-37. MR 81h:15015
- [P2]
- -, On the convex hull of the multiform numerical range, Linear and Multilinear Algebra 37 (1994), 221-223. MR 95j:15022
- [Ta]
- O. Taussky, Positive definite matrices, Inequalities I (O. Shisha, eds.), Academic Press, New York, 1965. MR 36:3806
- [Th]
- R. C. Thompson, Singular values, diagonal elements and convexity, SIAM J. Appl. Math. 32 (1977), 39-63. MR 54:12805
- [To]
- O. Toeplitz, Das algebraishe Analogon zu einem Satze von Fejér, Math. Z. 2 (1918), 187-197.
- [W]
- R. Westwick, A theorem on numerical range, Linear and Multilinear Algebra 2 (1975), 311-315. MR 51:11132
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(1991):
15A60,
47A12
Retrieve articles in all Journals with MSC
(1991):
15A60,
47A12
Additional Information:
Yiu
Tung
Poon
Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email:
ytpoon@iastate.edu
DOI:
10.1090/S0002-9939-97-03781-7
PII:
S 0002-9939(97)03781-7
Keywords:
Generalized numerical range,
convexity,
joint positive definiteness,
multiple eigenvalue
Received by editor(s):
September 22, 1995
Received by editor(s) in revised form:
January 4, 1996
Additional Notes:
The author wants to thank the referee for some helpful comments and suggestions.
Dedicated:
Dedicated to Professor Yik Hoi Au-Yeung
Communicated by:
Palle E. T. Jorgensen
Copyright of article:
Copyright
1997,
American Mathematical Society
|
Comments: webmaster@ams.org