Polynomial numerical hulls of order 3

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Let A be any n-by-n normal matrix and let k > 0 be an integer. By using the concept of the joint numerical range W (A, A 2 , · · · , A k ), an analytic description of V k (A) for normal matrices will be presented. Additionally, new proof for Theorem 2.2 of Davis-Li-Salemi [Linear Algebra Appl., 428 (2008), pp. 137-153] is given.

2009

Received by the editors December 28, 2008. Accepted for publication April 17, 2009. Handling Editor: Bit-Shun Tam. Department of Mathematics, Vali-E-Asr University of Rafsanjan, Rafsanjan, Iran ([email protected], [email protected]). Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran, and Center of Excellence in Linear Algebra and Optimization of Shahid Bahonar University of Kerman, Iran ([email protected]). Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 18, pp. 253-263, May 2009

2013

In this note we characterize polynomial numerical hulls of matrices A 2 Mn such that A 2 is Hermitian. Also, we consider normal matrices A 2 Mn whose k th power are semidenite. For such matrices we show that V k (A) = (A).

Let J k (λ) be the k × k Jordan block with eigenvalue λ and let N be an m × m normal matrix. In this paper we study the polynomial numerical hulls of order 2 and n − 1 for A = J k (λ) ⊕ N , where n = m + k. We obtain a necessary and sufficient condition such that V 2 (A) has an interior point. Also, we analytically characterize V 2 (J 2 (λ) ⊕ N) and we show that if σ(N) ∪ {λ} is co-linear, then V 2 (J 2 (λ) ⊕ N) = a∈σ(N) V 2 (J 2 (λ) ⊕ [a]). Finally, we study V n−1 (A) and we show that if σ(N) is neither co-linear nor co-circular, then V n−1 (A) has at most one point more than σ(A).

In this paper, the behavior of the pseudopolynomial numerical hull of a square complex matrix with respect to struc-tured perturbations and its radius is investigated.

Applied Mathematics and Computation, 2006

Let A be an n • n matrix. In this paper we discuss theoretical properties of the polynomial numerical hull of A of degree one and assemble them into three algorithms to computing the numerical range of A.

Linear Algebra and its Applications, 2006

Six characterizations of the polynomial numerical hull of degree k are established for bounded linear operators on a Hilbert space. It is shown how these characterizations provide a natural distinction between interior and boundary points. One of the characterizations is used to prove that the polynomial numerical hull of any fixed degree k for a Toeplitz matrix whose symbol is piecewise continuous approaches all or most of that of the infinitedimensional Toeplitz operator, as the matrix size goes to infinity.

Linear Algebra and its Applications, 2019

Let J n (λ) be the n × n Jordan block with a positive real eigenvalue λ and A := J n (λ) ⊕ J n (−λ). In this paper, we study the polynomial numerical hull of degree k, H k (A), when k = 2 and k = 2n − 1. If 0 ≤ λ ≤ 3 2 , we show that [−λ − r 2,n , λ + r 2,n ] ⊆ H 2 (A), where r 2,n represents the radius of the circular disk H 2 (J n (0)) and if λ > √ 2, then there exists α > 0 such that H 2 (A) ∩{x + iy ∈ C : x 2 − y 2 < α} = ∅. Also, we give explicit formulas for computing the sets H 3 (A) and H 2 (A) ∩R, when n = 2. Although the spectrum and numerical range are continuous set valued functions with respect to the Hausdorff metric, by an example, we show that polynomial numerical hulls may not be even lower semicontinuous.

Linear Algebra and its Applications, 2018

The joint numerical range W (F) of three hermitian 3-by-3 matrices F = (F 1 , F 2 , F 3) is a convex and compact subset in R 3. We show that W (F) is generically a three-dimensional oval. Assuming dim(W (F)) = 3, every one-or two-dimensional face of W (F) is a segment or a filled ellipse. We prove that only ten configurations of these segments and ellipses are possible. We identify a triple F for each class and illustrate W (F) using random matrices and dual varieties.

Linear Algebra and its Applications, 1997

Some algebraic properties of the sharp points of the numerical range of matrix polynomials are the main subject of this paper. We also consider isolated points of the numerical range and the location of the numerical range in a circular annulus. 0 1997 Elsevier Science Inc.