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Numerical ranges and matrix completions

Hadwin, D.W., Harrison, K.J. and Ward, J.A. (2000) Numerical ranges and matrix completions. Linear Algebra and its Applications, 315 (1-3). pp. 145-154.

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There are two natural ways of defining the numerical range of a partial matrix. We show that for each partial matrix supported on a given pattern they give the same convex subset of the complex plane if and only if a graph associated with the pattern is chordal. This extends a previously known result (C.R. Johnson, M.E. Lundquist, Operator Theory: Adv. Appl. 50 (1991) 283–291) to patterns that are not necessarily reflexive and symmetric, and our proof overcomes an apparent gap in the proof given in the above-mentioned reference. We also define a stronger completion property that we show is equivalent to the pattern being an equivalence.

Publication Type: Journal Article
Murdoch Affiliation: School of Mathematical and Physical Sciences
Publisher: Elsevier
Copyright: 2000 Elsevier Science Inc.
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