On matrices whose numerical ranges have circular or weak circular symmetry
✍ Scribed by Bit-Shun Tam; Shangjun Yang
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 205 KB
- Volume
- 302-303
- Category
- Article
- ISSN
- 0024-3795
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✦ Synopsis
In , among other equivalent conditions, it is proved that a square complex matrix A is permutationally similar to a block-shift matrix if and only if for any complex matrix B with the same zero pattern as A, W (B), the numerical range of B, is a circular disk centered at the origin. In this paper, we add a long list of further new equivalent conditions. The corresponding result for the numerical range of a square complex matrix to be invariant under a rotation about the origin through an angle of 2π/m, where m 2 is a given positive integer, is also proved. Many interesting by-products are obtained. In particular, on the numerical range of a square nonnegative matrix A, the following unexpected results are established: (i) when the undirected graph of A is connected, if W (A) is a circular disk centered at the origin, then so is W (B), for any complex matrix B with the same zero pattern as A; (ii) when A is irreducible, if λ is an eigenvalue in the peripheral spectrum of A that lies on the boundary of W (A), then λ is a sharp point of W (A). We also obtain results on the numerical range of an irreducible square nonnegative matrix, which strengthen or clarify the work of Issos [9] and Nylen and Tam [14] on this topic.