zbMATH — the first resource for mathematics

On the generalized numerical range. (English) Zbl 0648.15017
Let $$A_ k$$, $$k=1,...,m$$, be $$n\times n$$ Hermitian matrices. Let $$f: {\mathbb{C}}$$ $$n\to {\mathbb{R}}^ m$$have components $$f\quad k(x)=x\quad HA_ kx,$$ $$k=1,...,m$$, $$W(A_ 1,...,A_ k)=\{f(x): \| x\| =1\}.$$ It is known that W is convex when $$n\geq 3$$ and $$m=3$$, and W is not convex in general when $$m>3$$. The authors give geometric proofs of these results and study the geometry of W.
Reviewer: N.M.Zobin

MSC:
 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
Full Text:
References:
 [1] Hausdorff F., Main. Z. 3 pp 314– (1919) [2] DOI: 10.1090/S0002-9939-1961-0122827-1 [3] DOI: 10.1080/03081088008817357 · Zbl 0437.15009 [4] DOI: 10.1007/BF01212904 · JFM 46.0157.02 [5] Friedland S., Pacific J. Math. 62 (1976) [6] Au-Yeung Y.-H., Southeast Asian Bull. Math. 3 pp 85– (1979) [7] DOI: 10.1090/S0002-9939-1983-0712625-4 [8] Doyle J. C., Proceedingsof IEE 129 pp 242– (1982) [9] Fan M. K. H., University of Maryland.College Park, Maryland, Technical Report TK-87-9, March 1987, submitted for publication. See also Proceedings of the American Control Conference pp 437– (1986) [10] Hauser John E., ElectronicsResearch Laboratory (1986) [11] DOI: 10.1307/mmj/1028997958 · Zbl 0082.11601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.