Perturbation thèorems for the generalized eigenvalue problem

Given a matrix pair Z = (A, B), the perturbation of its eigenvalues (α, β) is studied. Considering two pairs Z, W as points of the Grassman manifold G_{n, 2n} and its eigenvalues as points in G_{1, 2}, the projective complex plane, the distance of the spectra, measured in the chordal metric in G_{1, 2}, is bounded by some distance of the matrix pairs in G_{n, 2n}. Analogs of the Bauer-Fike theorem, Henrici's theorem, and the Hoffman-Weilandt theorem are obtained, from which the “classical” results can be derived.     

The eigenvalue distribution on Schur complements of H-matrices

The paper studies the eigenvalue distribution of some special matrices. Tong in Theorem 1.2 of [Wen-ting Tong, On the distribution of eigenvalues of some matrices, Acta Math. Sinica (China), 20 (4) (1977) 273-275] gives conditions for an n × n matrix A ∈ SD_n ∪ ID_n to have |J_R+(A)| eigenvalues with positive real part, and |J_R-(A)| eigenvalues with negative real part. A counter-example is given in this paper to show that the conditions of the theorem are not true. A corrected condition is then proposed under which the conclusion of the theorem holds. Then the corrected condition is applied to establish some results about the eigenvalue distribution of the Schur complements of H-matrices with complex diagonal entries. Several conditions on the n × n matrix A and the subset α ⊆ N = {1, 2, … , n} are presented such that the Schur complement matrix A/α of the matrix A has eigenvalues with positive real part and eigenvalues with negative real part.     

The finite spectrum of Sturm-Liouville problems with transmission conditions

We study the finite spectrum of Sturm-Liouville problems with transmission conditions. For any positive integer n, we construct a class of regular Sturm-Liouville problems with transmission conditions, which have exactly n eigenvalues, and these n eigenvalues can be located anywhere in the complex plane in non-self-adjoint case and anywhere along the real line in the self-adjoint case.     

Contractive maps on normed linear spaces and their applications to nonlinear matrix equations

In this paper we will give necessary and sufficient conditions under which a map is a contraction on a certain subset of a normed linear space. These conditions are already well known for maps on intervals in R. Using the conditions and Banach’s fixed point theorem we can prove a fixed point theorem for operators on a normed linear space. The fixed point theorem will be applied to the matrix equation X = I_n + A^∗f(X)A, where f is a map on the set of positive definite matrices induced by a real valued map on (0, ∞). This will give conditions on A and f under which the equation has a unique solution in a certain set. We will consider two examples of f in detail. In one example the application of the fixed point theorem is the first step in proving that the equation has a unique positive definite solution under the conditions on A.     

A complete unitary similarity invariant for unicellular matrices

We present necessary and sufficient conditions for an n×n complex matrix B to be unitarily similar to a fixed unicellular (i.e., indecomposable by similarity) n×n complex matrix A.     

Weyl's theorem for closed operators on Fréchet spaces

A celebrated theorem of H. Weyl asserts that if A is a normal operator on a Hilbert space X, then the points in the spectrum of A which can be removed by perturbing A with a compact operator are precisely the eigenvalues of finite multiplicity which are isolated points of the spectrum of A. In these notes we develop appropriate Fredholm theory that enables us to provide interesting sufficiency and necessary conditions for a closed operator on a Fréchet space to satisfy Weyl's theorem. We complement and extend results of L. A. Coburn, V. Istr??escu, S. K. Berberian, and M. Schechter, respectively.     

Krein Orthogonal Entire Matrix Functions and Related Lyapunov Equations: A State Space Approach

For a class of entire matrix valued functions of exponential type new necessary and sufficient conditions are derived in order that these functions are Krein orthogonal functions. The conditions are stated in terms of certain operator Lyapunov equations. These equations arise by using infinite dimensional state space representations of the entire matrix functions involved. As a corollary, using a recent operator inertia theorem, we give a new proof of the Ellis-Gohberg-Lay theorem which relates the number of zeros of a Krein orthogonal function in the open upper half plane to the number of negative eigenvalues of the corresponding selfadjoint convolution operator.     

Nondifferentiable boundary points of the higher numerical range

Let A be an n-square complex matrix. Every nondifferentiable point on ?W_m(A), the boundary of the mth numerical range of A, is a sum of m eigenvalues of A. This generalizes a theorem of W. F. Donoghue. Moreover, if sufficiently many sums of m eigenvalues of A occur on ?W_m(A), then A is normal. From these results it follows that if ?W_m(A) is a convex polygon with sufficiently many vertices, then A is normal.     

Extension of continuous functions in digital spaces with the Khalimsky topology

The digital space Zn equipped with Efim Khalimsky's topology is a connected space. We study continuous functions Zn⊃A→Z, from a subset of Khalimsky n-space to the Khalimsky line. We give necessary and sufficient condition for such a function to be extendable to a continuous function Zn→Z. We classify the subsets A of the digital plane such that every continuous function A→Z can be extended to a continuous function on the whole plane.     

On the reduction of pairs of hermitian or symmetric matrices to diagonal form by congruence

Let A₁, A₂ be given n-by-n Hermitian or symmetric matrices, and consider the simultaneous transformations A_i→SA_iS^* if A_i is Hermitian or A_i→SA_iS^T if A_i is symmetric. We give necessary and sufficient conditions for the existence of a unitary S which reduces both A₁ and A₂ to diagonal form in this way. When at least one of A₁ or A₂ is nonsingular, we give necessary and sufficient conditions for a reduction of this sort with a nonsingular S. These results are a generalization of the classical theorem from mechanics that a positive definite matrix and a Hermitian matrix can always be diagonalized simultaneously by a nonsingular congruence.     

Congruence of Hermitian matrices by Hermitian matrices

Two Hermitian matrices A,B∈M_n(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C∈M_n(C) such that B=CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible inertias of the Hermitian matrices C that carry the congruence. We also give necessary and sufficient conditions for any 2-by-2 nonsingular Hermitian matrices to be Hermitian-congruent. In both of the studied cases, we show that if A and B are real and Hermitian-congruent, then they are congruent by a real symmetric matrix. Finally we note that if A and B are 2-by-2 nonsingular real symmetric matrices having the same sign pattern, then there is always a real symmetric matrix C satisfying B=CAC. Moreover, if both matrices are positive, then C can be picked with arbitrary inertia.     

Drazin-monotonicity characterizations for property-n matrices

A square matrix A is said to have property n if there exists a nonnegative power of A. In this paper, necessary and sufficient conditions for such matrices to have a nonnegative Drazin inverse are presented.     

Inertia theorems for matrices,controllability, and linear vibrations

If H is a Hermitian matrix and $\text{W = AH + HA}{}^1$ is positive definite, then A has as many eigenvalues with positive (negative) real part as H has positive (negative) eigenvalues [5]. Theorems of this type are known as inertia theorems. In this note the rank of the controllability matrix of A and W is used to derive a new inertia theorem. As an application, a result in [8] and [4] on a damping problem of the equation $\text{M}\text{x}\text{?}\text{+ (D + G) xdot; + Kx = 0}$ is extended.     

Sums of Laplace eigenvalues—rotationally symmetric maximizers in the plane

The sum of the first n?1 eigenvalues of the Laplacian is shown to be maximal among triangles for the equilateral triangle, maximal among parallelograms for the square, and maximal among ellipses for the disk, provided the ratio ³(area)/(moment of inertia) for the domain is fixed. This result holds for both Dirichlet and Neumann eigenvalues, and similar conclusions are derived for Robin boundary conditions and Schrödinger eigenvalues of potentials that grow at infinity. A key ingredient in the method is the tight frame property of the roots of unity. For general convex plane domains, the disk is conjectured to maximize sums of Neumann eigenvalues.     

The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law

LetAbe annbynmatrix whose elements are independent random variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized eigenvalues is asymptotically uniform in the unit disk in the complex plane. We derive the exact expected empirical spectral distribution of the complex eigenvalues for finiten, from which convergence in the expected distribution to the circular law for normally distributed matrices may be derived. Similar methodology allows us to derive a joint distribution formula for the real Schur decomposition ofA. Integration of this distribution yields the probability thatAhas exactlykreal eigenvalues. For example, we show that the probability thatAhas all real eigenvalues is exactly 2^{−n(n−1)/4}.     

Approximation of isolated eigenvalues of general singular ordinary differential operators

Let A be a self-adjoint operator defined by a general singular ordinary differential expression τ on an interval (a, b), ? ∞ ≤ a < b ≤ ∞. We show that isolated eigenvalues in any gap of the essential spectrum of A are exactly the limits of eigenvalues of suitably chosen self-adjoint realizations A_n of τ on subintervals (a_n, b_n) of (a, b) with a_n → a, b_n → b. This means that eigenvalues of singular ordinary differential operators can be approximated by eigenvalues of regular operators. In the course of the proof we extend a result, which is well known for quasiregular differential expressions, to the general case: If the spectrum of A is not the whole real line, then the boundary conditions needed to define A can be given using solutions of (τ ? λ)u = 0, where λ is contained in the regularity domain of the minimal operator corresponding to τ.     

On fixed point sets of homeomorphisms of then-ball

Conditions are investigated under which a subsetA can be the fixed point set of a homeomorphism ofB ⁿ . If eitherA ∩ ?B ⁿ ≠ Ø andn arbitrary orA ∩ ?B ⁿ =Ø andn even it is necessary and sufficient thatA is non-empty and closed. IfA ∩ ?B ⁿ =Ø andn odd, conditions which are either necessary or sufficient (but not both) are given.     

Birkhoff's theorem and multidimensional numerical range

We show that, under certain conditions, Birkhoff's theorem on doubly stochastic matrices remains valid for countable families of discrete probability spaces which have nonempty intersections. Using this result, we study the relation between the spectrum of a self-adjoint operator A and its multidimensional numerical range. It turns out that the multidimensional numerical range is a convex set whose extreme points are sequences of eigenvalues of the operator A. Every collection of eigenvalues which can be obtained by the Rayleigh-Ritz formula generates an extreme point of the multidimensional numerical range. However, it may also have other extreme points.     

Some angularity and inertia theorems related to normal matrices

Motivated by the definition of the inertia, introduced by Ostrowski and Schneider, a notion of angularity of a matrix is defined. The angularity characterizes the distribution of arguments of eigenvalues of a matrix. It is proved that if B and C are nonsingular matrices, then B¹AB and C¹AC have the same angularity provided they are diagonal. Some well-known inertia theorems (e.g. Sylvester's law) have been deduced as corollaries of this result. The case when C is permitted to be singular is discussed next. Finally, we prove that (a) any linear transformation T, on the set of n by n complex matrices, mapping Hermitian matrices into themselves and preserving the inertia of each Hermitian matrix is of the form T(A)=C¹AC or T(A)=C^1LA′C where C is some nonsingular matrix, and (b) any linear transformation T mapping normal matrices into normal matrices and preserving the angularity of each normal matrix is also of one of the above forms, but with C=kU where k≠0 and U is unitary.     

Generalized exponential dichotomy and global linearization

Hartman's linearization theorem says that if all eigenvalues of matrix A have no zero real part and f(x) is small Lipschitzian, then nonlinear system x^′=Ax+f(x) and its linear system x^′=Ax are topologically equivalent. In 1970s Palmer extended the theorem to nonautonomous systems. In this paper we extend Hartman's theorem to the systems with generalized exponential dichotomy.