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Abstract
An n-by-n () weighted shift matrix A is one of the form
where the
’s, called the weights of A, are complex numbers. Let
denote the
-by-
principal submatrix of A obtained by deleting its jth row and jth column. We show that the boundary of numerical range W(A) has a line segment if and only if the
’s are nonzero and
for some
. This refines previous results of Tsai and Wu on numerical ranges of weighted shift matrices. In addition, we give an example showing that there is a weighted shift matrix with line segments on the boundary of its numerical range such that the moduli of its weights are not periodic.
Keywords:
AMS Subject Classification:
Acknowledgments
We thank the (anonymous) referee for his helpful suggestions and pointing out the general result of Lemma 7 to us, which led to considerable improvements in the exposition. The research was supported by the National Science Council of the Republic of China under NSC-100-2115-M-008-004. Dedicated to Professor Pei YuanWu.