Polynomials and numerical ranges

Li, Chi-Kwong

doi:10.1090/S0002-9939-1988-0962800-3

Polynomials and numerical ranges
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Proc. Amer. Math. Soc. 104 (1988), 369-373 Request permission

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.

References

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