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Graphs can be associated with a matrix according to some rule and we can find the spectrum of a graph with respect to that matrix. Two graphs are cospectral if they have the same spectrum. Constructions of cospectral graphs help us…
It is known that the generating function $f$ of a sequence of Toeplitz matrices $\{T_n(f)\}_n$ may not describe the asymptotic distribution of the eigenvalues of $T_n(f)$ if $f$ is not real. In a recent paper, we assume as a working…
Motivated by interpretability and reliability, we investigate whether large language models (LLMs) deploy universal geometric structures to encode discrete, graph-structured knowledge. To this end, we present two complementary experimental…
We investigate the universality of singular value and eigenvalue distributions of matrix valued functions of independent random matrices and apply these general results in several examples. In particular we determine the limit distribution…
We prove, under a certain representation theoretic assumption, that the set of real symmetric matrices, whose eigenvalues satisfy a linear matrix inequality, is itself a spectrahedron. The main application is that derivative relaxations of…
The authors analyze the asymptotics of eigenvalues of Toeplitz matrices with certain continuous and discontinuous symbols. In particular, the authors prove a conjecture of Levitin and Shargorodsky on the near-periodicity of Toeplitz…
Spectra of ordered eigenvalues of finite Random Matrices are interpreted as a time series. Dataadaptive techniques from signal analysis are applied to decompose the spectrum in clearly differentiated trend and fluctuation modes, avoiding…
The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space $\mathcal{H}$ with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators…
We present a spectral theory of hypergraphs that closely parallels Spectral Graph Theory. A number of recent developments building upon classical work has led to a rich understanding of "hyperdeterminants" of hypermatrices, a.k.a.…
For a large class of symmetric random matrices with correlated entries, selected from stationary random fields of centered and square integrable variables, we show that the limiting distribution of eigenvalue counting measure always exists…
We consider $n\times n$ non-Hermitian random matrices with independent entries and a variance profile, as well as an additive deterministic diagonal deformation. We show that their empirical eigenvalue distribution converges to a limiting…
We develop the method of similar operators to study the spectral properties of unbounded perturbed linear operators that can be represented by matrices of various kinds. The class of operators under consideration includes various…
We study the spectral properties of a class of random matrices where the matrix elements depend exponentially on the distance between uniformly and randomly distributed points. This model arises naturally in various physical contexts, such…
We consider the probability of having two intervals (gaps) without eigenvalues in the bulk scaling limit of the Gaussian Unitary Ensemble of random matrices. We describe uniform asymptotics for the transition between a single large gap and…
We consider a non-relativistic quantum particle interacting with a singular potential supported by two parallel straight lines in the plane. We locate the essential spectrum under the hypothesis that the interaction asymptotically…
We analyze statistical properties of complex eigenvalues of random matrices $\hat{A}$ close to unitary. Such matrices appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with…
We consider a class of random banded Hessenberg matrices with independent entries having identical distributions along diagonals. The distributions may be different for entries belonging to different diagonals. For a sequence of $n\times n$…
We formulate gaussian and circular random-matrix models representing a coupled system consisting of an absorbing and an amplifying resonator, which are mutually related by a generalized time-reversal symmetry. Motivated by optical…
We study scalar multivariate non-stationary subdivision schemes with a general dilation matrix. We characterize the capability of such schemes to reproduce exponential polynomials in terms of simple algebraic conditions on their symbols.…
With an increasing influx of classical signal processing methodologies into the field of graph signal processing, approaches grounded in discrete linear canonical transform have found application in graph signals. In this paper, we…