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A symmetric numerical range for matrices

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Summary

For each normv on ℂn, we define a numerical rangeZ v, which is symmetric in the sense thatZ v=ZvD, wherev D is the dual norm.

We prove that, fora ɛ ℂnn,Z v(a) contains the classical field of valuesV(a). In the special case thatv is thel 1-norm,Z v(a) is contained in a setG(a) of Gershgorin type defined by C. R. Johnson.

Whena is in the complex linear span of both the Hermitians and thev-Hermitians, thenZ v(a),V(a) and the convex hull of the usualv-numerical rangeV v(a) all coincide. We prove some results concerning points ofV(a) which are extreme points ofZ v(a).

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Authors and Affiliations

  1. Mathematical Sciences Department, Rensselaer Polytechnic Inst., 12181, Troy, NY, U.S.A.

    B. David Saunders

  2. Mathematics Department, University of Wisconsin, 53706, Madison, WI, U.S.A.

    Hans Schneider

Authors
  1. B. David Saunders
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  2. Hans Schneider
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David Saunders, B., Schneider, H. A symmetric numerical range for matrices. Numer. Math. 26, 99–105 (1976). https://doi.org/10.1007/BF01396569

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