Hermitian Matrix-valued Function Research Articles

On an eigenvector-dependent nonlinear eigenvalue problem from the perspective of relative perturbation theory

We are concerned with the eigenvector-dependent nonlinear eigenvalue problem (NEPv) H(V)V=VΛ, where H(V)∈ℂn×n is a Hermitian matrix-valued function of V∈ℂn×k with orthonormal columns, i.e., VHV=Ik, k≤n (usually k≪n). Sufficient conditions on the solvability and solution uniqueness of NEPv are obtained, based on the well-known results from the relative perturbation theory. These results are complementary to recent ones in Cai et al. (2018), where, among others, one can find conditions for the solvability and solution uniqueness of NEPv, based on the well-known results from the absolute perturbation theory. Although the absolute perturbation theory is more versatile in applications, there are cases where the relative perturbation theory produces better results.

Homogenization of the Fourth-Order Elliptic Operator with Periodic Coefficients with Correctors Taken into Account

An elliptic fourth-order differential operator $$A_\varepsilon$$ on $$L_2(\mathbb{R}^d;\mathbb{C}^n)$$ is studied. Here $$\varepsilon >0$$ is a small parameter. It is assumed that the operator is given in the factorized form $$A_\varepsilon = b(\mathbf{D})^* g(\mathbf{x}/\varepsilon) b(\mathbf{D})$$ , where $$g(\mathbf{x})$$ is a Hermitian matrix-valued function periodic with respect to some lattice and $$b(\mathbf{D})$$ is a matrix second-order differential operator. We make assumptions ensuring that the operator $$A_\varepsilon$$ is strongly elliptic. The following approximation for the resolvent $$(A_\varepsilon + I)^{-1}$$ in the operator norm of $$L_2(\mathbb{R}^d;\mathbb{C}^n)$$ is obtained: $$(A_{\varepsilon}+I)^{-1}=(A^{0}+I)^{-1}+\varepsilon K_{1}+\varepsilon^{2} K_{2}(\varepsilon)+O(\varepsilon^{3}).$$ Here $$A^0$$ is the effective operator with constant coefficients and $$K_{1}$$ and $$K_{2}(\varepsilon)$$ are certain correctors.

A Subspace Method for Large-Scale Eigenvalue Optimization

We consider the minimization or maximization of the $J$th largest eigenvalue of an analytic and Hermitian matrix-valued function, and build on Mengi, Yildirim, and Kilic [SIAM J. Matrix Anal. Appl., 35, pp. 699--724, 2014]. This work addresses the setting when the matrix-valued function involved is very large. We describe subspace procedures that convert the original problem into a small-scale one by means of orthogonal projections and restrictions to certain subspaces, and that gradually expand these subspaces based on the optimal solutions of small-scale problems. Global convergence and superlinear rate-of-convergence results with respect to the dimensions of the subspaces are presented in the infinite dimensional setting, where the matrix-valued function is replaced by a compact operator depending on parameters. In practice, it suffices to solve eigenvalue optimization problems involving matrices with sizes on the scale of tens, instead of the original problem involving matrices with sizes on the scale...

A Support Function Based Algorithm for Optimization with Eigenvalue Constraints

Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalue functions and is of practical interest because of a wide range of applications in fields such as structural design and control theory. Here we focus on the optimization of a linear objective subject to a constraint on the smallest eigenvalue of an analytic and Hermitian matrix-valued function. We propose a numerical approach based on quadratic support functions that overestimate the smallest eigenvalue function globally. The quadratic support functions are derived by employing variational properties of the smallest eigenvalue function over a set of Hermitian matrices. We establish the local convergence of the algorithm under mild assumptions and deduce a precise rate of convergence result by viewing the algorithm as a fixed point iteration. The convergence analysis reveals that the algorithm is immune to the nonsmooth nature of the smallest eigenvalue. We illustrate the ...

Solutions of a constrained Hermitian matrix-valued function optimization problem with applications

Abstract. Let f(X) = (XC + D )M(XC + D) − G be a given nonlinear Hermitian matrix-valued function with M = M∗ and G = G∗, and assume that the variable matrix X satisfies the consistent linear matrix equation XA = B. This paper shows how to characterize the semi-definiteness of f(X) subject to all solutions of XA = B. As applications, a standard method is obtained for finding analytical solutions X0 of X0A = B such that the matrix inequality f(X) < f(X0) or f(X) 4 f(X0) holds for all solutions of XA = B. The whole work provides direct access, as a standard example, to a very simple algebraic treatment of the constrained Hermitian matrix-valued function and the corresponding semi-definiteness and optimization problems.

An inverse problem for gyroscopic systems

A “gyroscopic system” is a Hermitian matrix-valued function of the form L(λ)=Mλ2+iGλ+C where M,G,C∈Rn×n with M>0 (positive definite), GT=−G≠0, CT=C and may be indefinite. Here we study factorizations of the form L(λ)=(Iλ−B)M(Iλ−A), where A,B∈Cn×n, and use them to construct gyroscopic systems with specified right divisor Iλ−A. We examine the constraints on the choice of A.

Some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix equation restrictions

Matrix rank and inertia optimization problems are a class of dis- continuous optimization problems in which the decision variables are matrices running over certain matrix sets, while the ranks and inertias of the vari- able matrices are taken as integer-valued objective functions. In this paper, we establish a group of explicit formulas for calculating the maximal and minimal values of the rank and inertia objective functions of the Hermitian matrix-valued function A1 B1XB 1 subject to the common Hermitian solu- tion of a pair of consistent matrix equations B2XB 2 = A2 and B3XB 3 = A3, and Hermitian solution of the consistent matrix equation B4X = A4, respec- tively. Many consequences are obtained, in particular, necessary and su- cient conditions are established for the triple matrix equations B1XB 1 = A1, B2XB 2 = A2 and B3XB 3 = A3 to have a common Hermitian solution, as well as necessary and sucient conditions for the two matrix equations B1XB 1 = A1 and B4X = A4 to have a common Hermitian solution.

Numerical Optimization of Eigenvalues of Hermitian Matrix Functions

This work concerns the global minimization of a prescribed eigenvalue or a weighted sum of prescribed eigenvalues of a Hermitian matrix-valued function depending on its parameters analytically in a box. We describe how the analytical properties of eigenvalue functions can be put into use to derive piece-wise quadratic functions that underestimate the eigenvalue functions. These piece-wise quadratic under-estimators lead us to a global minimization algorithm, originally due to Breiman and Cutler. We prove the global convergence of the algorithm, and show that it can be effectively used for the minimization of extreme eigenvalues, e.g., the largest eigenvalue or the sum of the largest specified number of eigenvalues. This is particularly facilitated by the analytical formulas for the first derivatives of eigenvalues, as well as analytical lower bounds on the second derivatives that can be deduced for extreme eigenvalue functions. The applications that we have in mind also include the ${\rm H}_\infty$-norm of a linear dynamical system, numerical radius, distance to uncontrollability and various other non-convex eigenvalue optimization problems, for which, generically, the eigenvalue function involved is simple at all points.

The Asymptotic Limits of Zero Modes of Massless Dirac Operators

Asymptotic behaviors of zero modes of the massless Dirac operator H = α · D + Q(x) are discussed, where α = (α1, α2, α3) is the triple of 4 × 4 Dirac matrices, $$D = \frac{1}{i } \nabla_x$$ , and Q(x) = (q jk (x)) is a 4 × 4 Hermitian matrix-valued function with | q jk (x) | ≤ C 〈x〉−ρ, ρ > 1. We shall show that for every zero mode f, the asymptotic limit of |x|2 f (x) as |x| → + ∞ exists. The limit is expressed in terms of the Dirac matrices and an integral of Q(x) f (x).

Asymptotic Spectra of Hermitian Block Toeplitz Matrices and Preconditioning Results

We study the asymptotic behavior of the eigenvalues of Hermitian n × n block Toeplitz matrices Tn, with k × k blocks, as n tends to infinity. No hypothesis is made concerning the structure of the blocks. Such matrices {Tn} are generated by the Fourier coefficients of a Hermitian matrix-valued function $f\in L^2$, and we study the distribution of their eigenvalues for large n, relating their behavior to some properties of f as a function; in particular, we show that the distribution of the eigenvalues converges to a limit $\mu_f$, and we explicitly compute $\mu_f$ in terms of f, showing that $\int F\, d\mu_f=1/k\int\tr F(f)$. Some consequences of this distribution and some localization results for the eigenvalues of Tn are discussed. We also study the eigenvalues of the preconditioned matrices {Pn-1Tn}, where the sequence {Pn} is generated by a positive definite matrix-valued function p. We show that the spectrum of any Pn-1Tn is contained in the interval [r,R], where r is the smallest and R the largest eigenvalue of p-1f. We also prove that the first m eigenvalues of Pn-1Tn tend to r and the last m tend to R, for any fixed m. Finally, the exact limit value of the condition number of the preconditioned matrices is computed.

Expansion of the one-loop effective action in covariant derivatives.

With an approach based on the heat-kernel representation, we show how to construct the expansion of the one-loop effective action in powers of covariant derivatives D/sub ..mu../ whenever it can be expressed in terms of an operator determinant of the form det(-D/sup 2/+V), where V is some positive Hermitian matrix-valued function. We present general expressions for the contributions to the effective Lagrangian in two and four covariant derivatives for four Euclidean space-time dimensions.