On polynomial numerical hulls of normal matrices

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.

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).

Electron. J. Linear Algebra, 2009

In this note, analytic description of V 3 (A) is given for normal matrices of the form

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.

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).

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, 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.

Linear Algebra and its Applications, 2002

The numerical range of an n × n matrix polynomial

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

Computers & Mathematics with Applications, 1996

Through the linearization of a matrix polynomial P(A), the symmetry and the sharp points of the numerical range w(P(A)) are studied.

2013

summary:Our aim is to classify and compute zeros of the quadratic two sided matrix polynomials, i.e. quadratic polynomials whose matrix coefficients are located at both sides of the powers of the matrix variable. We suppose that there are no multiple terms of the same degree in the polynomial $\mathbf{p}$, i.e., the terms have the form ${\mathbf{A}}_j{\mathbf{X}}^j{\mathbf{B}}_j$, where all quantities ${\mathbf{X}},{\mathbf{A}}_j,{\mathbf{B}}_j,j=0,1,\ldots,N,$ are square matrices of the same size. Both for classification and computation, the essential tool is the description of the polynomial $\mathbf{p}$ by a matrix equation $\mathbf{P}(\mathbf{X}) := \mathbf{A}(\mathbf{X})\mathbf{X}+\mathbf{B}(\mathbf{X})$, where $\mathbf{A}(\mathbf{X})$ is determined by the coefficients of the given polynomial $\mathbf{p}$ and $\mathbf{P}, \mathbf{X},\mathbf{B}$ are real column vectors. This representation allows us to classify five types of zero points of the polynomial $\mathbf{p}$ in dependen...

Linear Algebra and its Applications, 1998

An investigation on nonconnectedness of numerical range for manic matrix polynomials L(1) is undertaking here. The distribution of eigenvalues of L(1) to the components of numerical range and some other algebraic properties are also presented.

Linear Algebra and its Applications, 2001

For any n × n matrices A and C, we consider the star-centers of three sets, namely, the C-numerical range W C (A) of A, the set diag U(A) of diagonals of matrices in the unitary orbit of A, and the set S(A) of matrices whose C-numerical ranges are contained in W C (A) for all C. For normal matrices A, we show that the set of star-centers of W A * (A) is a bounded closed real interval, and give complete description of the sets of star-centers of diag U(A) and of S(A). In particular, we show that if A is normal with noncollinear eigenvalues, then each of S(A) and diag U(A) has exactly one star-center. For general square matrices A, we also give sufficient conditions for the sets of star-centers of diag U(A) and of S(A) to be singleton sets.

Applied Mathematics and Computation, 2010

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some respects, then it is close to be multiple, and we construct an upper bound for this distance (measured in the euclidean norm). We also derive a new expression for the condition number of a simple eigenvalue, which does not involve eigenvectors. Moreover, an Elsner-like perturbation bound for matrix polynomials is presented. j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a m c