Volume 42, pp. 1-12, 2014.

Revisiting the inverse field of values problem

Natália Bebiano, João da Providência, Ana Nata, and João P. da Providência


The field of values of a linear operator is the convex set in the complex plane comprising all Rayleigh quotients. For a given complex matrix, Uhlig proposed the inverse field of values problem: given a point inside the field of values, determine a unit vector for which this point is the corresponding Rayleigh quotient. In the present note we propose an alternative method of solution to those that have appeared in the literature. Our approach is based on the fact that the field of values can be seen as a union of ellipses under a compression to the two-dimensional case, in which case the problem has an exact solution. Refining an idea of Marcus and Pesce, we provide alternative algorithms to plot the field of values of a general complex matrix, which perform faster and more accurately than the existing ones.

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Key words

field of values, numerical range, inverse problem, generating vector, compression

AMS subject classifications

15A60, 47B35

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