The numerical range of selfadjoint matrix polynomials

Matrix polynomials of the form $P(\lambda)=I\lambda^2+A_1\lambda+A_0$ (where A0 and A1 are n × n Hermitian matrices and $\lambda$ is a complex variable) arise in many applications. The numerical range of such a polynomial is \[ W(P) = \{\lambda \in \mathbb{C} : x^*P(\lambda)x = 0 \mbox{ for some nonzero } x \in \mathbb{C}^n \} \] and it always contains the spectrum of $P(\lambda)$, i.e., the set of zeros of $\textup{det}P(\lambda)$. Properties of the numerical range are developed in detail, taking advantage of the close connection between W(P) and the classical numerical range (field of values) of the (general) complex matrix A:=A0+iA1. Eigenvalues and nondifferentiable points on the boundary are examined and a procedure for the numerical determination of W(P) is presented and used for several illustrations. Some extensions of the theory to more general polynomials $P(\lambda)$ are also discussed, as well as special cases describing vibrating systems.

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.

Componentwise analysis of direct factorization of real symmetric and hermitian matrices

Many natural phenomena can be modeled by a second-order dynamical system , where stands for an appropriate state variable and M, C, K are time-invariant, real and symmetric matrices. In contrast to the classical inverse vibration problem where a model is to be determined from natural frequencies corresponding to various boundary conditions, the inverse mode problem concerns the reconstruction of the coefficient matrices (M, C, K) from a prescribed or observed subset of natural modes. This paper set forth a mathematical framework for the inverse mode problem and resolves some open questions raised in the literature. In particular, it shows that given merely the desirable structure of the spectrum, namely given the cardinalities of real or complex eigenvalues but not of the actual eigenvalues, the set of eigenvectors can be completed via solving an under-determined nonlinear system of equations. This completion suffices to construct symmetric coefficient matrices (M, C, K) whereas the underlying system can have arbitrary eigenvalues. Generic conditions under which the real symmetric quadratic inverse mode problem is solvable are discussed. Applications to important tasks such as updating models without spill-over or constructing models with positive semi-definite coefficient matrices are discussed.

The inertia-preservers of several sets of matrices are identified. The sets include: all real matrices, all complex matrices, triangular matrices, real symmetric matrices and Hermitian matrices.

We consider the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a real symmetric random matrix. Our main result is that the existing result for a random matrix from the Gaussian Orthogonal Ensemble, obtained by Brézin and Hikami (2001), essentially continues to hold for a general real symmetric Wigner matrix. To obtain this result, we adapt the approach by Götze and Kösters (2008), who proved the analogous result for the Hermitian case.

Let Gr,p be a graph obtained from a path by adjoining a cycle Cr of length r to one end and the central vertex of a star Sp on p vertices to the other end.In this paper, it is proven that unicyclic graph Gr,p with r even is determined by its Laplacian spectrum except for n = p + 4.

Various topics of matrix theory, in particular, those related to nonnegative matrices (matrices with nonnegative entries) are considered in this chapter. We introduce the concepts of reducible and irreducible matrices and matrix graph theory (the concepts of directed and strongly connected graphs), and show the equivalence between irreducibility of a matrix and the connectivity of its directed graph. This enables us, among other things, to strengthen the Gershgorin theorem for estimating the location of eigenvalues of irreducible matrices. In order to determine if a symmetric matrix is positive definite, we need information regarding the signs of its eigenvalues. Also, in order to determine the rate of convergence of certain iterative methods to solve linear systems of algebraic equations, we need to know—as we shall see in later chapters—some information regarding the location of the eigenvalues of the iteration matrix. The Perron-Frobenius theorem, showing that the spectral radius ρ( A ) is an eigenvalue corresponding to a positive eigenvector, if A is nonnegative and irreducible, is presented. It will be seen in some of the following chapters that the concept of numerical radius can give sharper estimates of the norm of the powers of matrices, for instance, than the spectral radius can. Some results relating the numerical radius with the norm of the matrix and with the spectral radius of the symmetric part of nonnegative matrices are presented.

The numerical range of a matrix, according to Kippenhahn, is determined by a hyperbolic determinantal form of linear Hermitian matrices associated to the matrix. On the other hand, using Riemann theta functions, Helton and Vinnikov confirmed that a hyperbolic form always admits a determinantal representation of linear real symmetric matrices. The Riemann matrix of the hyperbolic curve plays the main role in the existence of real symmetric matrices. In this article, we implement computations of the Riemann matrix and the Abel–Jacobi variety of the hyperbolic curve associated to a determinantal polynomial of a matrix. Further, we prove that the lattice of the Abel–Jacobi variety is decomposed into the direct sum of two orthogonal lattices for some 4×4 Toeplitz matrices.

Two sufficient conditions under which the roots of the determinant of a given ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m \times m</tex> ) matrix polynomial of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> th order lie in the open left-half plane have been obtained. The first condition is given in terms of the positive definiteness of an ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mn \times mn</tex> ) symmetric matrix, while the second condition is given in terms of the positive definiteness of an ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m \times m</tex> ) matrix that is a function of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> , Re <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s \leq 0</tex> . These conditions are represented in terms of rational functions of the coefficient matrices of the given matrix polynomial. Therefore, the explicit computation of the determinant polynomial is not required.

THE PURPOSE of this paper is to develop relations among factors of raw scores, deviation scores, and double-centered scores. The treat ment is abstract and is merely an application of the algebra to the specialized matrices in which the interest of the factor analyst centers. Equiv alence of matrices, either square or rectangular, and similarity of square matrices are general notions that underlie the discussion. In addition, use is made of a specialized case of equivalence of matrices that occurs if the symmetric matrices generated by these matrices are elements of the same multiplicative group. Finally, a loose relation among matrices that generate symmetric matrices of different ranks belonging to the same sub-ring, or sub-algebra, of the total matric ring is employed to describe some of the results. It will be assumed that only real score matrices are employed and consequently that the symmetric ma trices they generate are not only real but also Gramian, that is, have no negative principal min ors. This is merely an arbitrary choice of fields, made because scores ordinarily are taken to be real numbers. The plan of the paper is first to describe a no tation that permits writing deviation scores as the matric product of raw scores and an idempo tent matrix. This notation should have a number of uses, a few of which are illustrated in this pa per. Next, the resolution of conventional devia tion scores into the product of factors and factor scores is summarized. This principle is, of course, well known; as it is summarized here, explicit use is made of the notion of groups o f singular matrices for which a symmetric idem potent matrix is a unit for multiplication. With the method established, it is possible to write equations relating the factors of raw scores to the factors of deviation scores, and equations re lating the factors of the two types of deviation scores. These results give a precise statement of Burt's reciprocity principle. Finally, double centered matrices are considered. Apparently two types of double-centered matrices may be identified; the relation of factors of such matrices to the factors of raw or deviation scores is out lined. It should be noted that an important restriction is made in this study. Throughout, it is assumed that the total variation exhibited in a set of data is to be analyzed in terms of common factors. This is a choice that the factor analyst may make. The results given here outline possibilities for relating solutions if he makes such a choice.

Low rank perturbations of strongly definitizable transformations and matrix polynomials

Thompson's famous theorems on singular values–diagonal elements of the orbit of an n×n matrix A under the action (1) U(n) ⊗ U(n) where A is complex, (2) SO(n) ⊗ SO(n), where A is real, (3) O(n) ⊗ O(n) where A is real are fully examined. Coupled with Kostant's result, the real semi-simple Lie algebra son, n yields (2) and hence (3) and the sufficient part (the hard part) of (1). In other words, the curious subtracted term(s) are well explained. Although the diagonal elements corresponding to (1) do not form a convex set in Cn, the projection of the diagonal elements into Rn (or iRn) is convex and the characterization of the projection is related to weak majorization. An elementary proof is given for this hidden convexity result. Equivalent statements in terms of the Hadamard product are also given. The real simple Lie algebra sun, n shows that such a convexity result fits into the framework of Kostant's result. Convexity properties and torus relations are studied. Thompson's results on the convex hull of matrices (complex or real) with prescribed singular values, as well as Hermitian matrices (real symmetric matrices) with prescribed eigenvalues, are generalized in the context of Lie theory. Also considered are the real simple Lie algebras sop, q and sop, q, p < q, which yield the rectangular cases. It is proved that the real part and the imaginary part of the diagonal elements of complex symmetric matrices with prescribed singular values are identical to a convex set in Rn and the characterization is related to weak majorization. The convex hull of complex symmetric matrices and the convex hull of complex skew symmetric matrices with prescribed singular values are given. Some questions are asked.

Given three sets of $n$ real eigenvalues satisfying the trace equality and the Horn inequalities, we know that there are $n\times n$ real symmetric matrices $A$ and $B$ so that $A$ has the first set of eigenvalues, $B$ has the second set of eigenvalues, and $A+B$ has the last set of eigenvalues. Under the condition that $B$ is a rank 2 matrix, we give a construction for the matrices $A$ and $B$. This construction is based on performing two orthogonal rank 1 updates on $A$. We end with a discussion of the relationship between this rank 2 Horn problem and the following similar problem: given a set of $n$ real eigenvalues, a set of $2$ real eigenvalues, and a set of $n+2$ real eigenvalues satisfying certain conditions, find an $(n+2)\times(n+2)$ real symmetric matrix such that the top left principal submatrix has the first set of eigenvalues, the bottom right principal submatrix has the second set of eigenvalues, and the full matrix has the last set of eigenvalues.

In the field of operator algebra, the concept of antieigenvalue theory was first introduced by Karl Gustafson with a special emphasis on accretive operators. In a wide range of scientific applications, antieigenvalue naturally occurs. Due to the inclusion of nonlinear Euler equations in the antieigenvalue theory, computing antieigenvalues is a difficult task as compared to that of computing eigenvalues of the operator. In the current paper, we consider linear two-parameter eigenvalue problems (LTEP) and will discuss the abstract algebraic setting of the problem as proposed by Atkinson. We analyze the generalized antieigenvalue pair and their corresponding generalized antieigenvectors for LTEP using the consequences of the Cauchy Schwarz inequality. Some generalized antieigenvalue bounds will also be derived. Generalized antieigenvalues and their corresponding generalized antieigenvectors will be calculated solving their relevant optimization problem. For numerical computations, three examples will be provided. Real symmetric matrices are used in the first case, while real diagonal matrices are used in the second case and finally arbitrary matrices are considered.