The Numerical Range of Self-Adjoint Quadratic Matrix Polynomials

The numerical range of derivations

The numerical range of a selfadjoint matrix polynomial \( L\left( \lambda \right) = \sum\nolimits_j^\ell { = 0{\lambda ^j}{A_j}} \) is the set of points μ∈ℂ for which x* L(μ)x = 0 for some nonzero vector x. As for the classical eigenvalue problem (when L(λ) = λ I − A), the spectrum of L(λ) is contained in its numerical range. Properties of the numerical range are investigated with special emphasis on the cases when L(λ) has only real spectrum (and, possibly, the point at infinity) and when the coefficients of the matrix polynomial are real symmetric matrices.

Quadratic realizability of palindromic matrix polynomials

In this paper, we study the stability of matrix polynomials under structured perturbations of their coefficients. More precisely, we consider a family of matrix polynomials \[ P_u(λ)=A_d(u)λ^d+A_{d-1}(u)λ^{d-1}+\cdots+A_0(u), \] whose matrix coefficients depend continuously and semialgebraically on a parameter vector $u\in\mathbb{C}^p$. Assuming that the matrix polynomial is monic, we show that the spectrum, the $\varepsilon$-pseudospectrum, the numerical range, and the joint numerical range associated with $P_u(λ)$ define set-valued maps that are Hölder continuous with respect to the parameter $u$. Moreover, the parameter space $\mathbb{C}^p$ can be decomposed into a finite union of analytic semialgebraic submanifolds such that, on each submanifold, the eigenvalues and the Jordan pairs of $P_u(λ)$ depend analytically on $u$. We also note that most of the results remain valid if the monicity assumption is replaced by the local nonsingularity of the leading coefficient matrix $A_d(u)$. However, the monic setting is adopted throughout the paper in order to simplify the exposition and to avoid additional technical assumptions, which are required in particular for results concerning numerical ranges.

The field of values (or numerical range) is an important notion in the generalization of facts about Hermitian matrices to general complex matrices. Completions of partial Hermitian matrices have now been studied in some depth, and it is time to study completions of more general complex partial matrices. Two natural definitions for the “field of values” of a square partial complex are suggested. The first “builds up” from the inside of any completion, or a specification of the free entries, of a given partial matrix, while the second “whittles down” from the classical field of values of any completion. In general, the resulting sets are different, though there is a universal containment. In this note we characterize those patterns of specified entries for which the two definitions are identical. Chordal graphs again play a key role, but in a somewhat different way from the classical positive definite Hermitian case.KeywordsUndirected GraphHermitian MatrixHermitian MatriceNumerical RangeChordal GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Using the notions of the numerical range, Schur complement and unitary equivalence, an eigenvalue inequality is obtained for a general complex matrix, giving rise to a region in the complex plane that contains its spectrum. This region is determined by a curve, generalizing and improving classical eigenvalue bounds obtained by the Hermitian and skew-Hermitian parts, as well as the numerical range of a matrix.

Numerical ranges and stability estimates

Multilinear techniques are used to characterize unitary matrices in terms of a generalized numerical range. This characterization is then applied to analyze the structure of all linear operators on matrices which preserve this numerical range. The results generalize V. J. Pellegrini's determination of all linear operators preserving the classical numerical range.

Low rank perturbations of strongly definitizable transformations and matrix polynomials

It is known that solving coupled matrix equations with complex matrices can be very difficult and it is sufficiently complicated. In this work, we propose two iterative algorithms based on the Conjugate Gradient method (CG) for finding the reflexive and Hermitian reflexive solutions of the coupled Sylvester-conjugate matrix equations(including Sylvester and Lyapunov matrix equations as special cases). The iterative algorithms can automatically judge the solvability of the matrix equations over the reflexive and Hermitian reflexive matrices, respectively. When the matrix equations are consistent over reflexive and Hermitian reflexive matrices, for any initial reflexive and Hermitian reflexive matrices, the iterative algorithms can obtain reflexive and Hermitian reflexive solutions within a finite number of iterations in the absence of roundoff errors, respectively. Finally, two numerical examples are presented to illustrate the proposed algorithms.

Linear operators on matrices: the invariance of the decomposable numerical range

The concept of essential numerical range of an operator was dened and studied by Stamp i and Williams in 1972. Researchers generalised this idea of essential numerical range to a group of operators to the joint essential numerical range. In this paper, we consider the jointessential numerical range and show that the properties of the classical numerical range such as compactness also hold for the joint essential numerical range. Further, we show that the joint essential spectrum is contained in the joint essential numerical range by looking at the boundary of the joint essential spectrum.

For the classical numerical radius f(A) of an n-by-n real or complex matrix A, we consider inequalities of the following two types. in which f and g are polynomials. In the latter case, the event in which f and g are simple powers is of special interest. We present a variety of particular results depending upon the dimension n and the classification of A or the polynomials. A number of natural questions remain open.

Let W(A) be the numerical range of an n × n complex matrix A. Using algebraic geometry technique and a result of Kippenhahn, Anderson showed that if W(A) is contained in a circular disk, and W(A) contains more than n boundary points of the circular disk, then W(A) equals to the circular disk, and the center of the circular disk will be an eigenvalue of the matrix A. Many researchers have reproved and refined the result of Anderson. Very recently, Wu unified these results and proved the following statements for a matrix A of the form . 1. If W(A) is contained in a circular disk 𝒟 centered at α, and W(A) contains at least m + 1 boundary points of 𝒟, then W(A) = 𝒟. 2. If W(A) contains a circular disk 𝒟 centered at α, and the boundary of W(A) contains at least m + 1 boundary points of 𝒟, then the boundary of W(A) contains a circular arc which is the boundary of 𝒟. 3. If the boundary of W(A) contains at least 2m + 1 boundary points of a circular disk 𝒟 centered at α, then 𝒟 ⊆ W(A). Moreover, under any one of the three conditions, α is an eigenvalue of A with algebraic multiplicity larger than its geometric multiplicity. The proofs of Wu utilized the Bézout's theorem, Riesz-Fejér theorem, etc. In this note, short and elementary proofs using only simple properties of polynomials and continuous functions are given to Wu's results. Furthermore, the results are extended to the higher numerical ranges of matrices.

An inverse indefinite numerical range problem