Search

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 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.

In this paper, we consider the eigenvalue variation for any perturbation of Hermitian matrices, and we obtain two perturbation bounds. The first bound always improves the existing bound, and the second bound also improves the existing one under a suitable condition. A simple example is given for comparing these bounds.

Given a Hermitian matrix pencil $L(z) = zA - B$ with only real eigenvalues that are either of positive or negative type, the distance to a nearest Hermitian pencil outside the class is considered with respect to a specified norm. These problems are considered in the setting of the Hermitian $\epsilon$-pseudospectra of $L(z)$ with a proposed homogeneous form of the definition of eigenvalue type playing an important role in the investigations. A significant outcome of this analysis is a bisection-type algorithm for computing the Crawford number of a definite pencil and a nearest Hermitian pencil that is not definite. Each step of the algorithm requires the computation of the smallest eigenvalue(s) of a positive definite matrix of the same size as the original definite pencil and corresponding eigenvector(s). The results may also be extended to compute the distance from a definitizable pencil to a nearest Hermitian pencil that is not definitizable and a smallest Hermitian perturbation that attains this dista...

The development of strong linearizations preserving whatever structure a matrix polynomial might possess has been a very active area of research in the last years, since such linearizations are the starting point of numerical algorithms for computing eigenvalues of structured matrix polynomials with the properties imposed by the considered structure. In this context, Hermitian matrix polynomials are one of the most important classes of matrix polynomials arising in applications and their real eigenvalues are of great interest. The sign characteristic is a set of signs attached to these real eigenvalues which is crucial for determining the behavior of systems described by Hermitian matrix polynomials and, therefore, it is desirable to develop linearizations that preserve the sign characteristic of these polynomials, but, at present, only one such linearization is known. In this paper, we present a complete characterization of all the Hermitian strong linearizations that preserve the sign characteristic of a given Hermitian matrix polynomial and identify several families of such linearizations that can be constructed very easily from the coefficients of the polynomial.

The spectral behavior of regular Hermitian matrix pencils is examined under certain structure-preserving rank-1 and rank-2 perturbations. Since Hermitian pencils have signs attached to real (and infinite) blocks in canonical form, it is not only the Jordan structure but also this so-called sign characteristic that needs to be examined under perturbation. The observed effects are as follows: Under a rank-1 or rank-2 perturbation, generically the largest one or two, respectively, Jordan blocks at each eigenvalue lambda are destroyed, and if lambda is an eigenvalue of the perturbation, also one new block of size one is created at lambda. If lambda is real (or infinite), additionally all signs at lambda but one or two, respectively, that correspond to the destroyed blocks, are preserved under perturbation. Also, if the potential new block of size one is real, its sign is in most cases prescribed to be the sign that is attached to the eigenvalue lambda in the perturbation.

In [C.R. Johnson, B. Kroschel, M. Omladic, Eigenvalue multiplicities in principal submatrices, Linear Algebra Appl. 390 (2004) 111-120] a result constraining the eigenvalues of principal submatrices of complex Hermitian matrices, based upon matroid theory, was given. Here we give a matricial proof of this result which also enables us to find a generalization of the original result.

The eigenproblems of the rank-one updates of the matrices have lots of applications in scientific computation and engineering such as the symmetric tridiagonal eigenproblems by the divide-andconquer method and Web search engine. Many researchers have well studied the algorithms for computing eigenvalues of Hermitian matrices updated by a rank-one matrix [1–6]. Recently, Ding and Zhou studied a spectral perturbation theorem for rank-one updated matrices of special structure in [7] and considered two applications. Cheng, Luo, and Li considered the bounds of the smallest and largest eigenvalues for rank-one modification of Hermitian matrices [8]. Eigenvalue bounds for perturbations of Hermitian matrices have been considered by Ipsen and Nadler in [9]. In this paper, we consider the bounds of the eigenvalues for rank-one modification of Hermitian matrices. The ideas of this paper were mainly motivated by one of [9]. We study the following form

The inexact Rayleigh quotient iteration (RQI) is used for computing the smallest eigenpair of a large Hermitian matrix. It’s shown in this paper that under the uniform positiveness condition a new convergence theorem of the inexact RQI is presented and proved by the nature of eigenvalues. All the results are verified and analyzed by numerical experiments.

Suppose that the eigenvalues of an Hermitian matrix A whose graph is a tree T are known, as well as the eigenvalues of the principal submatrix of A corresponding to a certain branch of T. A method for constructing a larger tree T ', in which the branch is ‘`duplicated’', and an Hermitian matrix A′ whose graph is T ' is described. The eigenvalues of A' are all of those of A, together with those corresponding to the branch, including multiplicities. This idea is applied (1) to give a solution to the inverse eigenvalue problem for stars, (2) to prove that the known diameter lower bound, for the minimum number of distinct eigenvalues among Hermitian matrices with a given graph, is best possible for trees of bounded diameter, and (3) to increase the list of trees for which all possible lists for the possible spectra are know. A generalization of the basic branch duplication method is presented.

The overlap Dirac operator in lattice QCD requires the computation of the sign function of a matrix. While this matrix is usually Hermitian, it becomes non-Hermitian in the presence of a quark chemical potential. We show how the action of the sign function of a non-Hermitian matrix on an arbitrary vector can be computed efficiently on large lattices by an iterative method. A Krylov subspace approximation based on the Arnoldi algorithm is described for the evaluation of a generic matrix function. The efficiency of the method is spoiled when the matrix has eigenvalues close to a function discontinuity. This is cured by adding a small number of critical eigenvectors to the Krylov subspace, for which we propose two different deflation schemes. The ensuing modified Arnoldi method is then applied to the sign function, which has a discontinuity along the imaginary axis. The numerical results clearly show the improved efficiency of the method. Our modification is particularly effective when the action of the sign function of the same matrix has to be computed many times on different vectors, e.g., if the overlap Dirac operator is inverted using an iterative method., Comment: 11 pages, 4 figures; minor changes, as published in Comput. Phys. Commun.

Among the possible multiplicity lists for the eigenvalues of Hermitian matrices whose graph is a tree we focus upon M2, the maximum value of the sum of the two largest multiplicities. The corresponding M1 is already understood. The notion of assignment (of eigenvalues to subtrees) is formalized and applied. Using these ideas, simple upper and lower bounds are given for M2 (in terms of simple graph theoretic parameters), cases of equality are indicated, and a combinatorial algorithm is given to compute M2 precisely. In the process, several techniques are developed that likely have more general uses.

We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally $C^{2}$ and locally $C^{3}$ function (see Theorem \ref{t:U.t1}). The proof as our previous proof in \cite{Pa-Sh:97} is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials. Rather, we obtain the $sin$-kernel as a unique solution of a certain non-linear integro-differential equation that follows from the determinant formulas for the correlation functions of the model. We also give a simplified and strengthened version of paper \cite{BPS:95} on the existence and properties of the limiting Normalized Counting Measure of eigenvalues. We use these results in the proof of universality and we believe that they are of independent interest.

In this paper, we propose a numerical method to verify for nearly multiple eigenvalues of a Hermitian matrix not being strictly multiple eigenvalues. From approximate eigenvalues computed, it seems to be difficult to distinguish whether they are strictly multiple eigenvalues or simple ones, and if they are very close each other, the verification method for simple eigenvalues may fail to enclose them separately, because of singularity of the system in the verification. There are several methods for enclosing multiple and nearly multiple eigenvalues (e.g., [Rump, Computational error bounds for multiple or nearly multiple eigenvalues, Linear Algebra Appl. 324 (2001) 209-226]), For such cases, there is no result to decide the enclosed eigenvalues are nearly multiple or strictly multiple, up to now. So, for enclosed eigenvalues, we propose a numerical method to separate nearly multiple eigenvalues.

Suppose k 1

For Hermitian matrices, whose graph is a given tree, the relationships among vertex degrees, multiple eigenvalues and the relative position of the underlying eigenvalue in the ordered spectrum are discussed in detail. In the process, certain aspects of special vertices, whose removal results in an increase in multiplicity are investigated.

We compute the large scale (macroscopic) correlations in ensembles of normal random matrices with an arbitrary measure and in ensembles of general non-Hermition matrices with a class of non-Gaussian measures. In both cases the eigenvalues are complex and in the large $N$ limit they occupy a domain in the complex plane. For the case when the support of eigenvalues is a connected compact domain, we compute two-, three- and four-point connected correlation functions in the first non-vanishing order in 1/N in a manner that the algorithm of computing higher correlations becomes clear. The correlation functions are expressed through the solution of the Dirichlet boundary problem in the domain complementary to the support of eigenvalues. The two-point correlation functions are shown to be universal in the sense that they depend only on the support of eigenvalues and are expressed through the Dirichlet Green function of its complement.

Knutson and Tao's work on the Horn conjectures used combinatorial invariants called hives and honeycombs to relate spectra of sums of Hermitian matrices to Littlewood-Richardson coefficients and problems in representation theory, but these relationships remained implicit. Here, let $M$ and $N$ be two $n ×n$ Hermitian matrices. We will show how to determine a hive $\mathcal{H}(M, N)={H_ijk}$ using linear algebra constructions from this matrix pair. With this construction, one may also define an explicit Littlewood-Richardson filling (enumerated by the Littlewood-Richardson coefficient $c_μν ^λ$ associated to the matrix pair). We then relate rotations of orthonormal bases of eigenvectors of $M$ and $N$ to deformations of honeycombs (and hives), which we interpret in terms of the structure of crystal graphs and Littelmann's path operators. We find that the crystal structure is determined \emphmore simply from the perspective of rotations than that of path operators.

In this note, we study some basic properties of generalized eigenvalues of a definite Hermitian matrix pair. In particular, we prove an interlacing theorem and a minimax theorem. We also obtain upper bounds for the variation of the generalized eigenvalues under perturbation. These results extend and improve those of R.-C. Li, J.-g. Sun, and G. W. Stewart on the topic.

Orthogonal polynomials are traditionally studied as special functions in mathematical theories such as in the Hilbert space theory, differential equations and asymptotics. In this chapter, a new purpose of the generalized Hermite polynomials will be discussed in detail. The Lax pair obtained from the generalized Hermite polynomials can be applied to formulate the eigenvalue densities in the Hermitian matrix models with a general potential. The Lax pair method then solves the eigenvalue density problems on multiple disjoint intervals, which are associated with scalar Riemann-Hilbert problems for multi-cuts. The string equation can be applied to derive the nonlinear algebraic relations between the parameters in the density models by reformulating the potential function in terms of the trace function of the coefficient matrix obtained from the Lax pair and using the Cayley-Hamilton theorem in linear algebra. The Lax pair method improves the traditional methods for solving the eigenvalue densities by reducing the complexities in finding the nonlinear relations, and the parameters are then well organized for further analyzing the free energy function to discuss the phase transition problems.

Suppose A is a hermitian matrix of order n . Let λ 1 ⩾ λ 2 ⩾ … ⩾ λ n and a 11 , …, a nn denote the eigenvalues and diagonal elements of A . If k n with Σ k i=1 λ i = Σ k i=1 a ii , what structure is imposed on the hermitian matrix? We prove that it must be block-diagonal.

We give the perturbation bounds for the eigenprojections of a Hermitian matrix H = GJG∗ , where G has full column rank and J is nonsingular, under the perturbations of the factor G . Our bounds hold, for example, when G is given with elementwise relative error. Our bounds contain relative gaps between the eigenvalues and may thus be much less pessimistic than the standard norm estimates.

We prove some inequalities involving the eigenvalues of an nxn Hermitian matrix and the eigenvalues of the (n−1)x(n−1) principal submatrices. We apply this inequality to generalize a known result on the numerical range to the lth numerical range. The method used yields an elegant proof of the converse to the interlacing theorem, which we include. A counterexample to the quardratic spread inequality conjectured by R. C. Thompson is also given.

A general J-Hermitian matrix A may not have a basis of eigenvectors. In this chapter we describe the reduction to a block-diagonal form by a similarity transformation S-1AS. We pay particular attention to the problem of the condition number of the transformation matrix S which is a key quantity in any numerical manipulation.

Here we briefly overview the general eigenvalue problem Sx = λT x with two Hermitian matrices S, T and show how to reduce it to the case of a single J-Hermitian matrix A.

For a Hermitian matrix H with nonsingular principal submatrix A, it is shown that the eigenvalues of the Moore-Penrose inverse of the Schur complement (H/A) of A in H interlace the eigenvalues of the Moore-Penrose inverse of H. Moreover, if H is semidefinite, it is shown that the eigenvalues of (H/A) interlace the eigenvalues of H.

We study symmetry breaking in $Z_2$ symmetric large $N$ matrix models. In the planar approximation for both the symmetric double-well $\phi^4$ model and the symmetric Penner model, we find there is an infinite family of broken symmetry solutions characterized by different sets of recursion coefficients $R_n$ and $S_n$ that all lead to identical free energies and eigenvalue densities. These solutions can be parameterized by an arbitrary angle $\theta(x)$, for each value of $x = n/N < 1$. In the double scaling limit, this class reduces to a smaller family of solutions with distinct free energies already at the torus level. For the double-well $\phi^4$ theory the double scaling string equations are parameterized by a conserved angular momentum parameter in the range $0 \le l < \infty$ and a single arbitrary $U(1)$ phase angle., Comment: 23 pages and 4 figures, Preprint No. CERN-TH.6611/92, Brown HET-863, HUTP -- 92/A035, LPTHE-Orsay: 92/29

Inequalities are proved connecting the eigenvalue spread of a Hermitian matix to the eigenvalue spreads of its collection of principal submatrices. An application is made to the numerical range of general matrices.

Let A be Hermitian and let the orthonormal columns of X span an approximate invariant subspace of X. Then the residual $R = AX - XM\, ( M = X^H AX )$ will be small. The theorems of this paper bound the distance of the spectrum of M from the spectrum of A in terms of appropriate norms of R.

This paper presents a novel algorithm for computing the eigenvector associated with either the largest or the smallest eigenvalue of a complex Hermitian matrix. This type of algorithm is required for direction of arrival (DOA) and frequency estimation. Necessary and sufficient conditions for convergence are proved, and simulations show the superior performance over traditional methods.

In this paper, a new method is discussed to derive the eigenvalue density in a Hermitian matrix model with a general potential. The density is considered on one interval or multiple disjoint intervals. The method is based on Lax pair theory and the Cayley–Hamilton theorem by studying the orthogonal polynomials associated with the Hermitian matrix model. It is obtained that the restriction conditions for the parameters in the density are connected to the discrete Painleve I equation, and the results are related to the scalar Riemann–Hilbert problem. Some special density functions are also discussed in association with the known results in this subject.

We study the variance and the Laplace transform of the probability law of linear eigenvalue statistics of unitary invariant Matrix Models of n-dimentional Hermitian matrices as n tends to infinity. Assuming that the test function of statistics is smooth enough and using the asymptotic formulas by Deift et al for orthogonal polynomials with varying weights, we show first that if the support of the Density of States of the model consists of two or more intervals, then in the global regime the variance of statistics is a quasiperiodic function of n generically in the potential, determining the model. We show next that the exponent of the Laplace transform of the probability law is not in general 1/2variance, as it should be if the Central Limit Theorem would be valid, and we find the asymptotic form of the Laplace transform of the probability law in certain cases.

An hermitian matrix pencilλA − B withA nonsingular is called strongly definitizable ifAp(A−1B) is positive definite for some polynomialp. We present three characterizations of strongly definitizable pencils, which generalize the classical results for definite pencils. They are, in particular, stably simultaneously diagonable. We also discuss this form of stability with respect to an open subset of the real line. Implications for some quadratic eigenvalue problems are included.

In this paper we consider the behaviour of a real eigenvalue of an analytic Hermitian matrix valued function under perturbation with a positive semidefinite analytic Hermitian matrix valued function. We extend previous results on perturbation with positive definite functions to the positive semidefinite case.

A particularly simple proof of the inequality of Thompson and Therianos concerning sums of eigenvalues of a hermitian matrix is presented.

It is shown that the eigenvalues Ei of a Hermitian matrix H with matrix elements Hij = ΣkAkijak, where Akij are known numbers and ak a set of parameters, can be exactly expanded as E i = Σ k ( ∂E i ∂a k )a k . This property is applied to the analysis of the optical spectra of transition metal ions in crystals proposed by L. Pueyo, M. Bermejo, and J. W. Richardson (J. Solid State Chem. 31, 217, 1980), and it is shown that this method represents the best fit of the Hamiltonian eigenvalues to the observed (or calculated) spectrum. Further advantages of using this property, in connection with the spectral analysis, are the minimization of the errors associated with the numerical approximations and a reduction in computer time. In the molecular orbital calculation of the optical or uv spectra of these systems, this linear expansion of the eigenvalues give a detailed interpretation of the improvements produced by refined calculations, such as those including configuration interaction. In particular, the changes in one-electron energy and in open-shell repulsion interactions associated with the refinement can be clearly and easily formulated. As examples, the computed spectra of CrF4−6 and CrF3−6 are discussed.

Let be a hermitian matrix in partitioned form. Let the eigenvalues of A, B, C be α1 ≥ … ≥ αa, β1 ≥ … ≥ βb, γ1 ≥ … ≥ γn, respectively. In this paper four classes of inequalities are proved comparing the αi. and βj with the γk. The simplest of these is: if the subscripts is, js satisfy 1 ≤ i1 < … < im ≤ α, 1 ≤ j1 < … < jm ≤ b.

The behaviour of real eigenvalues of selfadjoint analytic matrix valued functions under small selfadjoint analytic perturbations is studied. Attention is paid mainly to the case when the perturbation is definite (or semidefi-nite). Earlier results of the authors concerning matrix polynomials of first degree are extended to the case of analytic functions.

In terms of a previously suggested Berger et al. (1977) iterative method for finding the lowest-lying eigenstate of hermitian matrixes the problem of pseudo-convergence, i.e. the tendency to converge to an undesired eigenvalue, is discussed. A simple extension of this algorithm is presented which provides a means of identifying and circumventing pseudo-convergence in many cases. This extended scheme may also be used simultaneously to obtain excited states in a numerically economical manner.

The goal of this article is to study how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations -- or in other words, to investigate optimal rigidity estimates for the eigenvalues. We do this in the setting of one-cut regular unitary invariant ensembles of random Hermitian matrices -- the Gaussian Unitary Ensemble being the prime example of such an ensemble. Our approach to this question combines extreme value theory of log-correlated stochastic processes, and in particular the theory of multiplicative chaos, with asymptotic analysis of large Hankel determinants with Fisher-Hartwig symbols of various types, such as merging jump singularities, size-dependent impurities, and jump singularities approaching the edge of the spectrum. In addition to optimal rigidity estimates, our approach sheds light on the fractal geometry of the eigenvalue counting function., 121 pages, 13 figures

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

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.

In this paper, we derive new relative perturbation bounds for eigenvectors and eigenvalues for regular quadratic eigenvalue problems of the form $\lambda^2 M x + \lambda C x + K x = 0$, where $M$ and $K$ are nonsingular Hermitian matrices and $C$ is a general Hermitian matrix. We base our findings on new results for an equivalent regular Hermitian matrix pair $A-\lambda B$. The new bounds can be applied to many interesting quadratic eigenvalue problems appearing in applications, such as mechanical models with indefinite damping. The quality of our bounds is demonstrated by several numerical experiments., Comment: 29 pages

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).

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.

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 $n \times n$ 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