Numerical ranges of companion matrices: flat portions on the boundary
Jeffrey Eldred, Leiba Rodman, Ilya M. Spitkovsky
TL;DR
This paper establishes criteria for companion matrices to have flat portions on the boundary of their numerical range, with specific results for 3x3 and 4x4 matrices, including impossibility results for certain cases.
Contribution
It provides new criteria for flat boundary portions of companion matrices' numerical ranges, including specialized results for small matrices and irreducibility constraints.
Findings
Criteria for flat portions on boundary of numerical range
Specialized results for 3x3 and 4x4 matrices
4x4 unitarily irreducible companion matrices cannot have 3 flat portions
Abstract
Criterion for a companion matrix to have a certain number of flat portions on the boundary of its numerical range is given. The criterion is specialized to the cases of 3-by-3 and 4-by-4 matrices. In the latter case, it is proved that a 4-by-4 unitarily irreducible companion matrix cannot have 3 flat portions on the boundary of its numerical range. Numerical examples are given to illustrate the main results.
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Taxonomy
TopicsMatrix Theory and Algorithms · Algebraic and Geometric Analysis · Holomorphic and Operator Theory