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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Polynomials and numerical ranges


Author: Chi-Kwong Li
Journal: Proc. Amer. Math. Soc. 104 (1988), 369-373
MSC: Primary 15A60; Secondary 15A69, 26C10
MathSciNet review: 962800
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Abstract: Let $ A$ be an $ n \times n$ complex matrix. For $ 1 \leq k \leq n$ we study the inclusion relation for the following polynomial sets related to the matrix $ A$.

(a) The classical numerical range of the $ k$th compound of the matrix $ \lambda I - A$.

(b) The $ k$th decomposable numerical range of the matrix $ \lambda I - A$.

(c) The convex hull of the set of all monic polynomials of degree $ k$ that divide the characteristic polynomial of $ A$. Moreover, we give an example showing that the set described in (a) is not convex in general. This settles a question raised by C. Johnson.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0962800-3
PII: S 0002-9939(1988)0962800-3
Keywords: Numerical range, decomposable numerical range, characteristic polynomial
Article copyright: © Copyright 1988 American Mathematical Society