![Sample our Mathematics & Statistics journals, sign in here to start your FREE access for 14 days]({"type":"image" , "src" : "/sda/5943/TOC-Subject-Sample-2021-Mathematics-Statistics.jpg"})
Abstract
We revisit an earlier idea of Marcus and Pesce from 1987 for generating fields of values of by
matrices
via 2 by 2 matrix compressions and their easily constructed elliptical field of values. This approach is used to reduce the cost of finding the FOV boundary curve of a matrix
and makes it more accurate at the same time. The new algorithm succeeds by using fewer eigenanalyses and constructing 2 by 2 matrix compression ellipses of
for approximating
instead of only eigenanalyses. An application to verify a counterexample to the
Zemanek Conjecture for companion matrices of monic polynomials and their normalized derivatives in
and
, respectively, in dimension 4 is given.
Acknowledgments
Some of the auxiliary functions for wberell2.m were extended from Haley Steger’s Master’s thesis at Auburn University (2012). To compare the accuracy of field of values plotters via their enclosed area was inspired by Tin-Yau Tam’s question after my talk on this subject at WONRA12 in Kaohsiung, Taiwan, in July 2012. The application to the companion matrix fields of values inclusion conjecture was suggested by the referee. I am very grateful for reminding me of this conjecture and for supplying the example polynomial. Finally, Pei Yuan Wu sent me his best reconstruction of the history of Zemanek’s Conjecture for which I am very grateful.