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Generalized eigenvector plays an essential role in the signal processing field. In this paper, we present a novel neural network learning algorithm for estimating the generalized eigenvector of a Hermitian matrix pencil. Differently from some traditional algorithms, which need to select the proper values of learning rates before using, the proposed algorithm does not need a learning rate and is very suitable for real applications. Through analyzing all of the equilibrium points, it is proven that if and only if the weight vector of the neural network is equal to the generalized eigenvector corresponding to the largest generalized eigenvalue of a Hermitian matrix pencil, the proposed algorithm reaches to convergence status. By using the deterministic discrete-time method, some convergence conditions, which can be satisfied with probability 1, are also obtained to guarantee its convergence. Simulation results show that the proposed algorithm has a fast convergence speed and good numerical stability. The real application demonstrates its effectiveness in tracking the optimal vector of beamforming.

We consider large non-Hermitian random matrices $X$ with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having $2+\epsilon$ derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [Rider, Silverstein 2006], or the distribution of the matrix elements needed to be Gaussian [Rider, Virag 2007], or at least match the Gaussian up to the first four moments [Tao, Vu 2016; Kopel 2015]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of $X$ with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian Motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices $X$ that are presented in the companion paper [Cipolloni, Erdős, Schroder 2019].

We introduce the exceptional topological insulator (ETI), a non-Hermitian topological state of matter that features exotic non-Hermitian surface states which can only exist within the three-dimensional topological bulk embedding. We show how this phase can evolve from a Weyl semimetal or Hermitian three-dimensional topological insulator close to criticality when quasiparticles acquire a finite lifetime. The ETI does not require any symmetry to be stabilized. It is characterized by a bulk energy point gap, and exhibits robust surface states that cover the bulk gap as a single sheet of complex eigenvalues or with a single exceptional point. The ETI can be induced universally in gapless solid-state systems, thereby setting a paradigm for non-Hermitian topological matter., 6+ pages, 3 figures; 21 pages Supplemental Material

In the block version of the classical two-sided Jacobi method for the Hermitian eigenvalue problem, the off-diagonal elements of iterated matrix A ( k ) converge to zero. However, this fact alone does not necessarily guarantee that A ( k ) converges to a fixed diagonal matrix. The same is true for the matrix of accumulated unitary transformations Q ( k ) . We prove that under certain assumptions A ( k ) indeed converges to a fixed diagonal matrix, whose diagonal elements are the eigenvalues of the input matrix A. Next it is shown that for a simple eigenvalue the corresponding column of Q ( k ) converges to the corresponding eigenvector. For a multiple eigenvalue or a cluster of eigenvalues, we prove that the orthogonal projectors constructed from the corresponding columns of Q ( k ) converge to the orthogonal projector onto the eigenspace corresponding to those eigenvalues. Moreover, the appropriate convergence bounds are obtained for all discussed cases. Convergence results are also valid for the parallel block-Jacobi method with dynamic ordering. The developed theory is illustrated by numerical example.

In this work, a class of inexact block SSOR-like preconditioners are proposed for the non-Hermitian positive definite linear system whose coefficient matrix is of a blockwise form. The novel preconditioners are based on the SSOR-like iteration method proposed by Bai (Numer. Linear Algebra Appl. 23, 37-60, 2016), but each of the diagonal blocks in the SSOR-like method is replaced by an approximation matrix which is easier to deal with. The estimated bounds for eigenvalues of the new preconditioned matrix, as well as the convergence property about the corresponding iteration method, are discussed. Finally, numerical experiments arise from the discretization of two-dimensional fractional diffusion equation and two-dimensional linear integro-differential equation are presented to confirm the theoretical analyses and illustrate the efficiency of the new preconditioners.

Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many applications. Linearizations of rational...

In this article, we propose the Fast Algorithms for Maxwell’s Equations (FAME) package for solving Maxwell’s equations for modeling three-dimensional photonic crystals. FAME combines the null-space free method with fast Fourier transform (FFT)-based matrix-vector multiplications to solve the generalized eigenvalue problems (GEPs) arising from Yee’s discretization. The GEPs are transformed into a null-space free standard eigenvalue problem with a Hermitian positive-definite coefficient matrix. The computation times for FFT-based matrix-vector multiplications with matrices of dimension 7 million are only 0.33 and 3.6 × 10 − 3 seconds using MATLAB with an Intel Xeon CPU and CUDA C++ programming with a single NVIDIA Tesla P100 GPU, respectively. Such multiplications significantly reduce the computational costs of the conjugate gradient method for solving linear systems. We successfully use FAME on a single P100 GPU to solve a set of GEPs with matrices of dimension more than 19 million, in 127 to 191 seconds per problem. These results demonstrate the potential of our proposed package to enable large-scale numerical simulations for novel physical discoveries and engineering applications of photonic crystals.

The discretizations of the left and the right fractional derivatives based on the shifted finite-difference formulas of the Grunwald-Letnikov type can result in Toeplitz matrices T and T ⁎ . Combining with the generating function of Toeplitz matrix T, we analyse the dominant property of the Hermitian part of T relative to its skew-Hermitian part. Then we construct a dominant Hermitian splitting iteration method for solving the discrete linear system of the considered space-fractional diffusion equations, and design a more practical dominant Hermitian-circulant splitting preconditioner to accelerate the convergence rates of the Krylov subspace iteration methods. Theoretical analyses demonstrate that all eigenvalues of the corresponding preconditioned matrix are clustered in a complex disk centered at 1 with the radius much less than 1, especially when the order β of the fractional derivative is close to 2. In addition, the numerical results show that the constructed preconditioner can effectively solve the discrete linear systems of one-dimensional and two-dimensional space-fractional diffusion equations.

Resonance via Pade (RVP) is an efficient method for calculating autoionization resonance states. It is based on the stabilization technique in which the basis set is scaled. The scaling can be uniform (i.e., all basis functions are scaled) or partial. Herein, we compare the two RVP scaling schemes for calculating an autoionization eigenvalue; moreover, the effect of freezing the core electrons is intertwined within this comparison. In order to study the different behavior of the RVP schemes, we associate each RVP scaling scheme with a complex contour of integration. Similarities between RVP and other non-Hermitian methods emerge from the generated contours, which suggest that RVP introduces similar outgoing boundary conditions as the complex scaling (CS), complex basis function (CBF), and reflection-free complex absorbing potential (RF-CAP) methods. A uniform-RVP contour, unlike a partial one, immediately penetrates the complex plane and influences the interaction region. Hence, uniform scaling within RVP destroys the description of the core electrons, as well as the description of the reference state, and yields less reliable results than partial scaling. The 1s22p3s1P autoionization state of Be, at the equation-of-motion coupled-cluster level, is used as our case study model.

The change in geometric multiplicity, associated with perturbing a particular pair of off-diagonal entries, in a combinatorially symmetric matrix is investigated. First, we focus upon a Hermitian matrix, with a general graph, when the perturbation is Hermitian. This generalizes prior work in case the graph is a tree; the possibilities are much richer. Then, we turn to general matrices over a field. In both cases, by classifying the incident vertices, all possibilities for the change in geometric multiplicity are identified. Then, examples are given to show that each possibility may actually occur. In some instance, the change in geometric multiplicity depends upon the numerical value of the perturbation, and it matters that both off-diagonal entries change. However, in most instances, the change is qualitatively determined.

Many physical problems require the spectral analysis of quadratic matrix polynomials $M\lambda^2+D\lambda +K$, $\lambda \in \mathbb{C}$, with $n \times n$ Hermitian matrix coefficients, $M,\;D,\;K$. In this largely expository paper, we present and discuss canonical forms for these polynomials under the action of both congruence and similarity transformations of a linearization and also $\lambda$-dependent unitary similarity transformations of the polynomial itself. Canonical structures for these processes are clarified, with no restrictions on eigenvalue multiplicities. Thus, we bring together two lines of attack: (a) analytic via direct reduction of the $n \times n$ system itself by $\lambda$-dependent unitary similarity and (b) algebraic via reduction of $2n \times 2n$ symmetric linearizations of the system by either congruence (Section 4) or similarity (Sections 5 and 6) transformations which are independent of the parameter $\lambda$. Some new results are brought to light in the process. Complete descriptions of associated canonical structures (over $\mathbb{R}$ and over $\mathbb{C}$) are provided -- including the two cases of real symmetric coefficients and complex Hermitian coefficients. These canonical structures include the so-called sign characteristic. This notion appears in the literature with different meanings depending on the choice of canonical form. These sign characteristics are studied here and connections between them are clarified. In particular, we consider which of the linearizations reproduce the (intrinsic) signs associated with the analytic (Rellich) theory (Sections 7 and 9).

In this paper, we study T-eigenvalues of third-order tensors. Definitions of the T-eigenvalues and Hermitian tensors are proposed. We present a commutative tensor family. We prove some T-eigenvalue inequalities for Hermitian tensors, including extensions of Weyl's theorem and Cauchy's interlacing theorem from the matrix case to the tensor case. Finally, we introduce the stability of the T-eigenvalues and study the Lyapunov equation for tensors.

Matrix---not only the arrays of numbers but also has been used as a tool for many calculations in various subjects. Its inverse, eigen values, eigen vectors are of great importance to know its characters. In this paper I have discussed about some new properties of Hermitian, Skew Hermitian matrices, diagonalisation, eigen values, eigen vectors, spectrum, which will open up a new horizon to the students of Mathematics. Also, in this paper I have authored two totally new theorems for students and researchers . SB’s Theorem 1 is on Normality of a block diagonal matrix and SB’s Theorem 2 is on Spectrum of eigen values. These ideas came to me in course of teaching. Hope, these two theorems will be of great help for the students of Physics and Chemistry as well.

We explore the regularity of the roots of Garding hyperbolic polynomials and real stable polynomials. As an application we obtain new regularity results of Sobolev type for the eigenvalues of Hermitian matrices and for the singular values of arbitrary matrices. These results are optimal among all Sobolev spaces., 13 pages; some details added and typos corrected, accepted for publication in Proc. Amer. Math. Soc

If $A$ is an $n \times n$ Hermitian matrix with eigenvalues $\lambda_1(A),\dots,\lambda_n(A)$ and $i,j = 1,\dots,n$, then the $j^{\mathrm{th}}$ component $v_{i,j}$ of a unit eigenvector $v_i$ associated to the eigenvalue $\lambda_i(A)$ is related to the eigenvalues $\lambda_1(M_j),\dots,\lambda_{n-1}(M_j)$ of the minor $M_j$ of $A$ formed by removing the $j^{\mathrm{th}}$ row and column by the formula $$ |v_{i,j}|^2\prod_{k=1;k\neq i}^{n}\left(\lambda_i(A)-\lambda_k(A)\right)=\prod_{k=1}^{n-1}\left(\lambda_i(A)-\lambda_k(M_j)\right)\,.$$ We refer to this identity as the \emph{eigenvector-eigenvalue identity} and show how this identity can also be used to extract the relative phases between the components of any given eigenvector. Despite the simple nature of this identity and the extremely mature state of development of linear algebra, this identity was not widely known until very recently. In this survey we describe the many times that this identity, or variants thereof, have been discovered and rediscovered in the literature (with the earliest precursor we know of appearing in 1834). We also provide a number of proofs and generalizations of the identity., Comment: 28 pages, 1 figure, phase discussion expanded, matches published version

Let A be an n × n complex matrix. The spread of the matrix A, denotes by sp(A), is the largest distance between any two eigenvalues of A. It was introduced by L. Mirsky. In the present work we give...

Let $A$ be an $n \times n$ positive definite Hermitian matrix with all eigenvalues between 1 and 2. We represent the permanent of $A$ as the integral of some explicit log-concave function on ${\Bbb R}^{2n}$. Consequently, there is a fully polynomial randomized approximation scheme (FPRAS) for the permanent of $A$., 5 pages

This paper describes a set of rational filtering algorithms to compute a few eigenvalues (and associated eigenvectors) of non-Hermitian matrix pencils. Our interest lies in computing eigenvalues lo...

For an eigenvalue lambda(0) of a Hermitian matrix A, the formula of Thompson and McEnteggert gives an explicit expression of the adjugate of lambda I-0 - A, Adj(lambda I-0 - A), in terms of eigenvectors of Afor lambda(0) and all its eigenvalues. In this paper Thompson-McEnteggert's formula is generalized to include any matrix with entries in an arbitrary field. In addition, for any nonsingular matrix A, a formula for the elementary divisors of Adj(A) is provided in terms of those of A. Finally, a generalization of the eigenvalue-eigenvector identity and three applications of the Thompson-McEnteggert's formula are presented. Partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES. Partially supported by "Ministerio de Economia, Industria y Competitividad (MINECO)" of Spain and "Fondo Europeo de Desarrollo Regional (FEDER)" of EU through grants MTM2017-83624-P and MTM2017-90682-REDT, and by UPV/EHU through grant GIU16/42.

For Hermitian and non-Hermitian Hamiltonian matrices H, we present the Schr¨odinger equation for qudit (spin-j system, N-level atom) with the state vector |ψ〉 in a new form of the linear eigenvalue equation for the matrix ℋ = (H ⊗ 1N) and the probability eigenvector |p〉 identified with quantum states in the probability representation of quantum mechanics. We discuss the possibility to experimentally detect the difference between the system states described by the solutions, corresponding to the Schrodinger equation with Hermitian and non-Hermitian Hamiltonians, by measuring the probabilities of artificial spin-1/2 projections m = ± 1/2, sets of which are identified with qudit states. We show that different symmetries of systems, including 𝒫𝒯 -symmetry and broken 𝒫𝒯 -symmetry, are determined by a set of N complex eigenvalues of the Hamiltonian matrix H.

We consider the complex eigenvalues of a Wishart type random matrix model $X=X_1 X_2^*$, where two rectangular complex Ginibre matrices $X_{1,2}$ of size $N\times (N+\nu)$ are correlated through a non-Hermiticity parameter $\tau\in[0,1]$. For general $\nu=O(N)$ and $\tau$ we obtain the global limiting density and its support, given by a shifted ellipse. It provides a non-Hermitian generalisation of the Marchenko-Pastur distribution, which is recovered at maximal correlation $X_1=X_2$ when $\tau=1$. The square root of the complex Wishart eigenvalues, corresponding to the non-zero complex eigenvalues of the Dirac matrix $\mathcal{D}=\begin{pmatrix} 0 & X_1 \\ X_2^* & 0 \end{pmatrix},$ are supported in a domain parametrised by a quartic equation. It displays a lemniscate type transition at a critical value $\tau_c,$ where the interior of the spectrum splits into two connected components. At multi-criticality we obtain the limiting local kernel given by the edge kernel of the Ginibre ensemble in squared variables. For the global statistics, we apply Frostman's equilibrium problem to the 2D Coulomb gas, whereas the local statistics follows from a saddle point analysis of the kernel of orthogonal Laguerre polynomials in the complex plane., Comment: 30 pages, 4 figures, v2: references added, typos corrected

In this work, we study the adiabatic phases and shortcut for a pseudo-Hermitian interacting spin-1/2 composite system in which one system is driven by a non-Hermitian magnetic field. The biorthonormal eigenbasis and the condition for real eigenvalues have been derived and discussed. By studying the Berry phases and the transitionless quantum driving (TQD) Hamiltonian, the roles of correlation between the subsystems and the non-Hermicity are discussed. The non-Hermitian dynamics and the Berry phases of the subsystem, as well as their influences on the TQD Hamiltonian are studied in both no correlation and large correlation cases. These results mean that we can modify the correlation between the subsystems to control the adiabatic shortcuts of the subsystems. These schemes can provide a feasible way for studying the adiabatic shortcuts of the composite system and the non-Hermitian system.

We prove that any Hermitian matrix whose trace is integer and all eigenvalues lie in the segment [1 + 1/(k − 3),k − 1 − 1/(k − 3)] can be represented as a sum of k orthogonal projections. For the sums of k orthogonal projections, it is shown that the ratio of the number of eigenvalues that do not exceed 1 to the number of eigenvalues that are not smaller than 1 (with regard for the multiplicities) is not greater than k − 1. We also present examples of Hermitian matrices that satisfy the indicated condition for the numbers of eigenvalues but, at the same time, cannot be decomposed into the sum of k orthogonal projections.

Critical systems represent physical boundaries between different phases of matter and have been intensely studied for their universality and rich physics. Yet, with the rise of non-Hermitian studies, fundamental concepts underpinning critical systems - like band gaps and locality - are increasingly called into question. This work uncovers a new class of criticality where eigenenergies and eigenstates of non-Hermitian lattice systems jump discontinuously across a critical point in the thermodynamic limit, unlike established critical scenarios with spectrum remaining continuous across a transition. Such critical behavior, dubbed the “critical non-Hermitian skin effect”, arises whenever subsystems with dissimilar non-reciprocal accumulations are coupled, however weakly. This indicates, as elaborated with the generalized Brillouin zone approach, that the thermodynamic and zero-coupling limits are not exchangeable, and that even a large system can be qualitatively different from its thermodynamic limit. Examples with anomalous scaling behavior are presented as manifestations of the critical non-Hermitian skin effect in finite-size systems. More spectacularly, topological in-gap modes can even be induced by changing the system size. We provide an explicit proposal for detecting the critical non-Hermitian skin effect in an RLC circuit setup, which also directly carries over to established setups in non-Hermitian optics and mechanics., In non-Hermitian systems, fundamental concepts like bandgaps and locality cannot be applied as in Hermitian systems. Here, the authors introduce a class of non-Hermitian critical scenarios where the eigenstates and energies jump discontinuously across a critical point, with anomalous scaling properties

We generalize von Neumann's well-known trace inequality, as well as related eigenvalue inequalities for hermitian matrices, to Schatten-class operators between complex Hilbert spaces of infinite dimension. To this end, we exploit some recent results on the $C$-numerical range of Schatten-class operators. For the readers' convenience, we sketched the proof of these results in the Appendix., 16 pages

By introducing a regularization matrix and an additional iteration parameter, a new class of regularized deteriorated positive-definite and skew-Hermitian splitting (RDPSS) preconditioners are proposed for generalized saddle point linear systems. Compared with the well-known Hermitian and skew-Hermitian splitting (HSS) preconditioner and the regularized HSS preconditioner (Bai, 2019) studied recently, the new RDPSS preconditioners have much better computing efficiency especially when the (1,1) block matrix is non-Hermitian. It is proved that the corresponding RDPSS stationary iteration method is unconditionally convergent. In addition, clustering property of the eigenvalues of the RDPSS preconditioned matrix is studied in detail. Two numerical experiments arising from the meshfree discretization of a static piezoelectric equation and the finite element discretization of the Navier–Stokes equation show the effectiveness of the new proposed preconditioners.

A new way of solving the spectral problem to describe electronic excitations is demonstrated for a hydrogen atom. The applied methodology is based formally on the idea of electronic excitation description without introducing boundary conditions to the eigenvalue problem for the square of the angular momentum operator. The eigenvalues of such an operator are considered as complex in general. As a result, the spectral problem for the Schrödinger equation becomes non-Hermitian with complex energy values. The imaginary part of the total energy helps to estimate the excitation lifetime within a unified scheme. The existence of the Stark shift of atomic energy levels and the collapse of the atomic spectra are confirmed.

This paper offers a review of numerical methods for computation of the eigenvalues of Hermitian matrices and the singular values of general and some classes of structured matrices. The focus is on the main principles behind the methods that guarantee high accuracy even in the cases that are ill-conditioned for the conventional methods. First, it is shown that a particular structure of the errors in a finite precision implementation of an algorithm allows for a much better measure of sensitivity and that computation with high accuracy is possible despite a large classical condition number. Such structured errors incurred by finite precision computation are in some algorithms e.g. entry-wise or column-wise small, which is much better than the usually considered errors that are in general small only when measured in the Frobenius matrix norm. Specially tailored perturbation theory for such structured perturbations of Hermitian matrices guarantees much better bounds for the relative errors in the computed eigenvalues. % Secondly, we review an unconventional approach to accurate computation of the singular values and eigenvalues of some notoriously ill-conditioned structured matrices, such as e.g. Cauchy, Vandermonde and Hankel matrices. The distinctive feature of accurate algorithms is using the intrinsic parameters that define such matrices to obtain a non-orthogonal factorization, such as the \textsf{LDU} factorization, and then computing the singular values of the product of thus computed factors. The state of the art software is discussed as well.

If A is a real $2n \times 2n$ positive definite matrix, then there exists a symplectic matrix M such that $M^TAM=\text {diag}(D, D),$ where D is a positive diagonal matrix with diagonal entries $d_1(A)\leqslant \cdots \leqslant d_n(A).$ We prove a maxmin principle for $d_k(A)$ akin to the classical Courant–Fisher–Weyl principle for Hermitian eigenvalues and use it to derive an analogue of the Weyl inequality $d_{i+j-1}(A+B)\geqslant d_i(A)+d_j(B).$

Bai et al. proposed the multistep Rayleigh quotient iteration (MRQI) as well as its inexact variant (IMRQI) in a recent work (Comput. Math. Appl. 77: 2396–2406, 2019). These methods can be used to effectively compute an eigenpair of a Hermitian matrix. The convergence theorems of these methods were established under two conditions imposed on the initial guesses for the target eigenvalue and eigenvector. In this paper, we show that these two conditions can be merged into a relaxed one, so the convergence conditions in these theorems can be weakened, and the resulting convergence theorems are applicable to a broad class of matrices. In addition, we give detailed discussions about the new convergence condition and the corresponding estimates of the convergence errors, leading to rigorous convergence theories for both the MRQI and the IMRQI.

The preconditioned modified Hermitian/skew-Hermitian splitting (PMHSS) iteration method and the corresponding preconditioning technique can achieve satisfactory results for solving optimal control problems governed by Poisson’s equation. We explore the feasibility of such a method and preconditioner for solving optimization problems constrained by the more complicated Stokes system. Theoretical results demonstrate that the PMHSS iteration method is convergent because the spectral radius of the iterative matrix is less than $\frac {\sqrt {2}}{2}$ . Additionally, the PMHSS preconditioner still clusters eigenvalues on a unitary segment. It guarantees that the convergence of the PMHSS iteration method and preconditioning is independent of not only discretizing mesh size, but also of the Tikhonov regularization parameter. A more effective preconditioner is proposed based on the PMHSS preconditioner. The proposed preconditioner avoids the inner iterations when solving saddle point systems appearing in the generalized residual equations. Furthermore, it is still convergent and maintains its independence of parameter and mesh size.

Elastography is a ill-posed inverse problem that aims at recovering the Lame and density of the domain of interest from finite number of observations, we consider as model the Helmoltz equation. We present an implementation for solving the Helmholtz inverse problem in two dimensions via an optimization problem based on Bayesian approach. In addition, the accuracy of the method is also investigated with respect to the amount of information taken from the generalized Hermitian Eigenvalue problem and by comparing the maximum a posterior estimate to the true parameter distribution in simulated experiments.

The Ritz upper bound to eigenvalues of Hermitian operators is essential for many applications in science. It is a staple of quantum chemistry and physics computations. The lower bound devised by Temple in 1928 [G. Temple, Proc. R. Soc. A Math. Phys. Eng. Sci. 119, 276–293 (1928)] is not, since it converges too slowly. The need for a good lower-bound theorem and algorithm cannot be overstated, since an upper bound alone is not sufficient for determining differences between eigenvalues such as tunneling splittings and spectral features. In this paper, after 90 y, we derive a generalization and improvement of Temple’s lower bound. Numerical examples based on implementation of the Lanczos tridiagonalization are provided for nontrivial lattice model Hamiltonians, exemplifying convergence over a range of 13 orders of magnitude. This lower bound is typically at least one order of magnitude better than Temple’s result. Its rate of convergence is comparable to that of the Ritz upper bound. It is not limited to ground states. These results complement Ritz’s upper bound and may turn the computation of lower bounds into a staple of eigenvalue and spectral problems in physics and chemistry.

The paper describes a heuristic algorithm for solving a generalized Hermitian eigenvalue problem fast. The algorithm searches a subspace for an approximate solution of the problem. If the approximate solution is unacceptable, the subspace is expanded to a larger one, and then, in the expanded subspace a possibly better approximated solution is computed. The algorithm iterates these two steps alternately. Thus, the speed of the convergence of the algorithm depends on how to generate a subspace. In this paper, we derive a Riccati equation whose solution can correct the approximate solution of a generalized Hermitian eigenvalue problem to the exact one. In other words, the solution of the eigenvalue problem can be found if a subspace is expanded by the solution of the Riccati equation. This is a feature the existing algorithms such as the Krylov subspace algorithm implemented in the MATLAB and the Jacobi–Davidson algorithm do not have. However, similar to solving the eigenvalue problem, solving the Riccati equation is time-consuming. We consider solving the Riccati equation with low accuracy and use its approximate solution to expand a subspace. The implementation of this heuristic algorithm is discussed so that the computational cost of the algorithm can be saved. Some experimental results show that the heuristic algorithm converges within fewer iterations and thus requires lesser computational time comparing with the existing algorithms.

When a projection of a symmetric or Hermitian matrix to the positive semidefinite cone is computed approximately (or to working precision on a computer), a natural question is to quantify its accuracy. A straightforward bound invoking standard eigenvalue perturbation theory (e.g. Davis-Kahan and Weyl bounds) suggests that the accuracy would be inversely proportional to the spectral gap, implying it can be poor in the presence of small eigenvalues. This work shows that a small gap is not a concern for projection onto the semidefinite cone, by deriving error bounds that are gap-independent.

Recently a novel family of eigensolvers, called spectral indicator methods (SIMs), was proposed. Given a region on the complex plane, SIMs first compute an indicator by the spectral projection. The indicator is used to test if the region contains eigenvalue(s). Then the region containing eigenvalues(s) is subdivided and tested. The procedure is repeated until the eigenvalues are identified within a specified precision. In this paper, using Cayley transformation and Krylov subspaces, a memory efficient multilevel eigensolver is proposed. The method uses less memory compared with the early versions of SIMs and is particularly suitable to compute many eigenvalues of large sparse (non-Hermitian) matrices. Several examples are presented for demonstration.

The study of the laws of nature has traditionally been pursued in the limit of isolated systems, where energy is conserved. This is not always a valid approximation, however, as the inclusion of features such as gain and loss, or periodic driving, qualitatively amends these laws. A contemporary frontier of metamaterial research is the challenge open systems pose to the characterization of topological matter1,2. Here, one of the most relied upon principles is the bulk–boundary correspondence (BBC), which intimately relates the surface states to the topological classification of the bulk3,4. The presence of gain and loss, in combination with the violation of reciprocity, has been predicted to affect this principle dramatically5,6. Here, we report the experimental observation of BBC violation in a non-reciprocal topolectric circuit7, which is also referred to as the non-Hermitian skin effect. The circuit admittance spectrum exhibits an unprecedented sensitivity to the presence of a boundary, displaying an extensive admittance mode localization despite a translationally invariant bulk. Intriguingly, we measure a non-local voltage response due to broken BBC. Depending on the a.c. current feed frequency, the voltage signal accumulates at the left or right boundary, and increases as a function of nodal distance to the current feed. Boundary-localized bulk eigenstates given by the non-Hermitian skin effect are observed in a non-reciprocal topological circuit. A fundamental revision of the bulk–boundary correspondence in an open system is required to understand the underlying physics.

Exceptional points (EPs), at which both eigenvalues and eigenvectors coalesce, are ubiquitous and unique features of non-Hermitian systems. Second-order EPs are by far the most studied due to their abundance, requiring only the tuning of two real parameters, which is less than the three parameters needed to generically find ordinary Hermitian eigenvalue degeneracies. Higher-order EPs generically require more fine-tuning, and are thus assumed to play a much less prominent role. Here, however, we illuminate how physically relevant symmetries make higher-order EPs dramatically more abundant and conceptually richer. More saliently, third-order EPs generically require only two real tuning parameters in the presence of either a parity-time (PT) symmetry or a generalized chiral symmetry. Remarkably, we find that these different symmetries yield topologically distinct types of EPs. We illustrate our findings in simple models, and show how third-order EPs with a generic $\sim k^{1/3}$ dispersion are protected by PT symmetry, while third-order EPs with a $\sim k^{1/2}$ dispersion are protected by the chiral symmetry emerging in non-Hermitian Lieb lattice models. More generally, we identify stable, weak, and fragile aspects of symmetry-protected higher-order EPs, and tease out their concomitant phenomenology., minor typos fixed

We demonstrate that crystal defects can act as a probe of intrinsic non-Hermitian topology. In particular, in point-gapped systems with periodic boundary conditions, a pair of dislocations may induce a non-Hermitian skin effect, where an extensive number of Hamiltonian eigenstates localize at only one of the two dislocations. An example of such a phase are two-dimensional systems exhibiting weak non-Hermitian topology, which are adiabatically related to a decoupled stack of Hatano-Nelson chains. Moreover, we show that strong two-dimensional point-gap topology may also result in a dislocation response, even when there is no skin effect present with open boundary conditions. For both cases, we directly relate their bulk topology to a stable dislocation non-Hermitian skin effect. Finally, and in stark contrast to the Hermitian case, we find that gapless non-Hermitian systems hosting bulk exceptional points also give rise to a well-localized dislocation response., 6 pages, 4 figures, supplement included, accepted manuscript

Non-Hermitian systems show a non-Hermitian skin effect, where the bulk states are localized at a boundary of the systems with open boundary conditions. In this paper, we study the dependence of the localization length of the eigenstates on a system size in a specific non-Hermitian model with a critical non-Hermitian skin effect, where the energy spectrum undergoes discontinuous transition in the thermodynamic limit. We analytically show that the eigenstates exhibit remarkable localization, known as scale-free localization, where the localization length is proportional to a system size. Our result gives theoretical support for the scale-free localization, which has been proposed only numerically in previous works.

Since the introduction of Dyson's Brownian motion in early 1960's, there have been a lot of developments in the investigation of stochastic processes on the space of Hermitian matrices. Their properties, especially, the properties of their eigenvalues have been studied in great details. In particular, the limiting behaviors of the eigenvalues are found when the dimension of the matrix space tends to infinity, which connects with random matrix theory. This survey reviews a selection of results on the eigenvalues of stochastic processes from the literature of the past three decades. For most recent variations of such processes, such as matrix-valued processes driven by fractional Brownian motion or Brownian sheet, the eigenvalues of them are also discussed in this survey. In the end, some open problems in the area are also proposed., 28 pages

The geometry of Hamiltonian's eigenstates is encoded in the quantum geometric tensor (QGT). It contains both the Berry curvature, central to the description of topological matter and the quantum metric. So far the full QGT has been measured only in Hermitian systems, where the role of the quantum metric is mostly shown to determine corrections to physical effects. On the contrary, in non-Hermitian systems, and in particular near exceptional points, the quantum metric is expected to diverge and to often play a dominant role, for example on the enhanced sensing and on wave packet dynamics. In this work, we report the first experimental measurement of the quantum metric in a non-Hermitian system. The specific platform under study is an organic microcavity with exciton-polariton eigenstates, which demonstrate exceptional points. We measure the quantum metric's divergence and we determine the scaling exponent $n=-1.01\pm0.08$, which is in agreement with theoretical predictions for the second-order exceptional points.

For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner-Dyson-Mehta universality conjecture for the last remaining universality type in the complex Hermitian class. Our analysis holds not only for exact cusps, but approximate cusps as well, where an extended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp for both symmetry classes. This result is also used in the companion paper [arXiv:1811.04055] where the cusp universality for real symmetric Wigner-type matrices is proven., Comment: 58 pages, 2 figures. Updated introduction and references

Given a collection λ _ = { λ 1 \underline {\lambda }=\{\lambda _1 , … \dots , λ n } \lambda _n\} of real numbers, there is a canonical probability distribution on the set of real symmetric or complex Hermitian matrices with eigenvalues λ 1 , … , λ n \lambda _1, \ldots , \lambda _n . In this paper, we study various features of random matrices with this distribution. Our main results show that under mild conditions, when n n is large, linear functionals of the entries of such random matrices have approximately Gaussian joint distributions. The results take the form of upper bounds on distances between multivariate distributions, which allows us also to consider the case when the number of linear functionals grows with n n . In the context of quantum mechanics, these results can be viewed as describing the joint probability distribution of the expectation values of a family of observables on a quantum system in a random mixed state. Other applications are given to spectral distributions of submatrices, the classical invariant ensembles, and to a probabilistic counterpart of the Schur–Horn theorem, relating eigenvalues and diagonal entries of Hermitian matrices.

The singular values of a matrix A are defined as the square roots of the eigenvalues of A ⁎ A , and the energy of A denoted by E ( A ) is the sum of its singular values. The energy of a graph G, E ( G ) , is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In this paper, we prove that if A is a Hermitian matrix with the block form A = ( B D D ⁎ C ) , then E ( A ) ≥ 2 E ( D ) . Also, we show that if G is a graph and H is a spanning subgraph of G such that E ( H ) is an edge cut of G, then E ( H ) ≤ E ( G ) , i.e., adding any number of edges to each part of a bipartite graph does not decrease its energy. Let G be a connected graph of order n and size m with the adjacency matrix A. It is well-known that if G is a bipartite graph, then E ( G ) ≥ 4 m + n ( n − 2 ) | det ⁡ ( A ) | 2 n . Here, we improve this result by showing that the inequality holds for all connected graphs of order at least 7. Furthermore, we improve a lower bound for E ( G ) given in Oboudi (2019) [14] .

Bulk–boundary correspondence, a guiding principle in topological matter, relates robust edge states to bulk topological invariants. Its validity, however, has so far been established only in closed systems. Recent theoretical studies indicate that this principle requires fundamental revisions for a wide range of open systems with effective non-Hermitian Hamiltonians. Therein, the intriguing localization of nominal bulk states at boundaries, known as the non-Hermitian skin effect, suggests a non-Bloch band theory in which non-Bloch topological invariants are defined in generalized Brillouin zones, leading to a general bulk–boundary correspondence beyond the conventional framework. Here, we experimentally observe this fundamental non-Hermitian bulk–boundary correspondence in discrete-time non-unitary quantum-walk dynamics of single photons. We demonstrate pronounced photon localizations near boundaries even in the absence of topological edge states, thus confirming the non-Hermitian skin effect. Facilitated by our experimental scheme of edge-state reconstruction, we directly measure topological edge states, which are in excellent agreement with the non-Bloch topological invariants. Our work unequivocally establishes the non-Hermitian bulk–boundary correspondence as a general principle underlying non-Hermitian topological systems and paves the way for a complete understanding of topological matter in open systems. Measurements of non-Hermitian photon dynamics show boundary-localized bulk eigenstates given by the non-Hermitian skin effect. A fundamental revision of the bulk–boundary correspondence in open systems is required to understand the underlying physics.

Improving upon results of Rudelson and Vershynin, we establish delocalization bounds for eigenvectors of independent-entry random matrices. In particular, we show that with high probability every eigenvector is delocalized, meaning any subset of its coordinates carries an appropriate proportion of its mass. Our results hold for random matrices with genuinely complex as well as real entries. In both cases, our bounds match numerical simulations, up to lower order terms, indicating the optimality of our results. As an application of our methods, we also establish delocalization bounds for normal vectors to random hyperplanes. The proofs of our main results rely on a least singular value bound for genuinely complex rectangular random matrices, which generalizes a previous bound due to the first author, and may be of independent interest.

In [10], Pickrell fully characterizes the unitarily invariant probability measures on infinite Hermitian matrices. An alternative proof of this classification is given by Olshanski and Vershik in [9], and in [3] Borodin and Olshanski deduce from this proof that under any of these invariant measures, the extreme eigenvalues of the minors, divided by the dimension, converge almost surely. In this paper, we prove that one also has a weak convergence for the eigenvectors, in a sense that is made precise. After mapping Hermitian to unitary matrices via the Cayley transform, our result extends a convergence proven in our paper with Maples and Nikeghbali [6], for which a coupling of the circular unitary ensemble of all dimensions is considered.

We investigate the eigenvalue curves of 1-parameter hermitean and general complex or real matrix flows $A(t)$ in light of their geometry and the uniform decomposability of $A(t)$ for all parameters $t$. The often misquoted and misapplied results by Hund and von Neumann and by Wigner for eigencurve crossings from the late 1920s are clarified for hermitean matrix flows $A(t) = (A(t))^*$. A conjecture on extending these results to general non-normal or non-hermitean 1-parameter matrix flows is formulated and investigated. An algorithm to compute the block dimensions of uniformly decomposable hermitean matrix flows is described and tested. The algorithm uses the ZNN method to compute the time-varying matrix eigenvalue curves of $A(t)$ for $t_o \leq t\leq t_f$. Similar efforts for general complex matrix flows are described. This extension leads to many new and open problems. Specifically, we point to the difficult relationship between the geometry of eigencurves for general complex matrix flows $A(t)$ and a general flow's decomposability into blockdiagonal form via one fixed unitary or general matrix similarity for all parameters $t$., Comment: 15 pages, 11 graphs

We study certain pairs of subspaces V and W of C^n we call supports that consist of eigenspaces of the eigenvalues ±‖M‖ of a minimal hermitian matrix M(‖M‖ ≤‖M+D‖ for all real diagonals D). For any pair of orthogonal subspaces we define a non negative invariant δ called the adequacy to measure how close they are to form a support and to detect one. This function δ is the minimum of another map F defined in a product of spheres of hermitian matrices. We study the gradient, Hessian and critical points of F in order to approximate δ. These results allow us to prove that the set of supports has interior points in the space of flag manifolds. Fil: Mendoza, Alberto. Universidad Simón Bolívar; Venezuela Fil: Recht, Lázaro. Universidad Simón Bolívar; Venezuela. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina Fil: Varela, Alejandro. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina