In the recent years the transmission eigenvalue problem has been extensively studied for non-absorbing media. In this paper we initiate the study of this problem for absorbing media. In particular we show that, in the case of absorbing media, transmission eigenvalues form a discrete set, exist for sufficiently small absorption and for spherically stratified media exist without this assumption. ...

A stronger result on the limiting distribution of the eigenvalues of random Hermitian matrices of the form A+XTX∗, originally studied in Marčenko and Pastur [4], is presented. Here, X (N×n), T (n×n), and A (N×N) are independent, with X containing i.i.d. entries having finite second moments, T is diagonal with real (diagonal) entries, A is Hermitian, and n/N → c > 0 as N → ∞. Under addtional ass...

We show a lower bound on mixing time for a non-reversible Markov chain in terms of its eigenvalues. This is used to show a bound on the real part of the complex-valued eigenvalues in terms of the realvalued eigenvalues of a related reversible chain, and likewise to bound the second largest magnitude eigenvalue. A myriad of Cheeger-like inequalities also follow for non-reversible chains, which e...

In this work we present a new restart technique for iterative projection methods for nonlinear eigenvalue problems admitting minmax characterization of their eigenvalues. Our technique makes use of the minmax induced local enumeration of the eigenvalues in the inner iteration. In contrast to global numbering which requires including all the previously computed eigenvectors in the search subspac...

A monic quadratic Hermitian matrix polynomial L(λ) can be factorized into a product of two linear matrix polynomials, say L(λ) = (Iλ−S)(Iλ−A). For the inverse problem of finding a quadratic matrix polynomial with prescribed spectral data (eigenvalues and eigenvectors) it is natural to prescribe a right solvent A and then determine compatible left solvents S. This problem is explored in the pres...

A closed analytical expression is derived for the joint distribution function of the real and the imaginary parts of the eigenenergies of the operator H = H0 − iWW+ for the one-channel case, where H0 is taken from the Poissonian or one of the Gaussian ensembles with universality index β, and where the squared moduli |wα |2 of the components of W are assumed to be χ2-distributed with universalit...

We consider the Newton iteration for computing the principal matrix pth root, which is rarely used in the application for the bad convergence and the poor stability. We analyze the convergence conditions. In particular it is proved that the method converges for any matrix A having eigenvalues with modulus less than 1 and with positive real part. Based on these results we provide a general algor...

We derive upper and lower bounds on the smallest and largest eigenvalues, respectively, of real symmetric Toeplitz matrices. The bounds are rst obtained for positive-deenite matrices and then extended to the general real symmetric case. Our bounds are computed as the roots of rational and polynomial approximations to spectral, or secular, equations. The decomposition of the spectrum into even a...

A spectral function on a formally real Jordan algebra is a real-valued function which depends only on the eigenvalues of its argument. One convenient way to create them is to start from a function f : R 7→ R which is symmetric in the components of its argument, and to define the function F (u) := f(λ(u)) where λ(u) is the vector of eigenvalues of u. In this paper, we show that this construction...

Let G = (V,E) be a graph without loops and multiple edges. Let n and m be the number of vertices and edges of G, respectively. Such a graph will be referred to as an (n,m)-graph. For v ∈ V (G), let d(v) be the degree of v. In this paper, we are concerned only with undirected simple graphs (loops and multiple edges are not allowed). Let G be a graph with n vertices and the adjacency matrix A(G)....