相关论文: A note on switching eigenvalues under small pertur…

Detecting anomalies in large sets of observations is crucial in various applications, such as epidemiological studies, gene expression studies, and systems monitoring. We consider settings where the units of interest result in multiple…

In this paper, we describe a new algorithm that approximates the extreme eigenvalue/eigenvector pairs of a symmetric matrix. The proposed algorithm can be viewed as an extension of the Jacobi eigenvalue method for symmetric matrices…

Identifying feature representations in language models is a central task in mechanistic interpretability. Several recent studies have made the observation that feature representations can be inferred in some cases from singular vectors of…

This paper considers the problems of detecting a change point and estimating the location in the correlation matrices of a sequence of high-dimensional vectors, where the dimension is large enough to be comparable to the sample size or even…

The analysis of diagonalizable matrices in terms of their so-called isospectral reduction represents a versatile approach to the underlying eigenvalue problem. Starting from a symmetry of the isospectral reduction, we show in the present…

The eigenvalues and eigenvectors of nonnormal matrices can be unstable under perturbations of their entries. This renders an obstacle to the analysis of numerical algorithms for non-Hermitian eigenvalue problems. A recent technique to…

This paper is concerned with computations of a few smaller eigenvalues (in absolute value) of a large extremely ill-conditioned matrix. It is shown that smaller eigenvalues can be accurately computed for a diagonally dominant matrix or a…

In a recent paper, wavelet analysis was used to perturb the coupling matrix in an array of identical chaotic systems in order to improve its synchronization. As the synchronization criterion is determined by the second smallest eigenvalue…

We derive new perturbation bounds for eigenvalues of Hermitian matrices with block structures. The structures we consider range from a standard 2-by-2 block form to block tridiagonal and tridigaonal forms. The main idea is the observation…

We consider a matrix pencil whose coefficients depend on a positive parameter $\epsilon$, and have asymptotic equivalents of the form $a\epsilon^A$ when $\epsilon$ goes to zero, where the leading coefficient $a$ is complex, and the leading…

We obtain general, exact formulas for the overlaps between the eigenvectors of large correlated random matrices, with additive or multiplicative noise. These results have potential applications in many different contexts, from quantum…

A remarkable mathematical property -- somehow hidden and recently rediscovered -- allows obtaining the eigenvectors of a Hermitian matrix directly from their eigenvalues. That opens the possibility to get the wavefunctions from the…

The term interlacing refers to systematic inequalities between the sequences of eigenvalues of two operators defined on objects related by a specific oper- ation. In particular, knowledge of the spectrum of one of the objects then implies…

Many complex systems can be reduced to their key components through spectrally decomposing matrices that capture their dynamics. These matrices can in turn be constructed from data, often by least-squares fitting: examples of algorithms to…

High-dimensional vector autoregression with measurement error is frequently encountered in a large variety of scientific and business applications. In this article, we study statistical inference of the transition matrix under this model.…

This paper proposes a novel methodology for the online detection of changepoints in the factor structure of large matrix time series. Our approach is based on the well-known fact that, in the presence of a changepoint, a factor model can be…

We consider an Information-Plus-Noise type matrix where the Information matrix is a spiked matrix. When some eigenvalues of the random matrix separate from the bulk, we study how the corresponding eigenvectors project onto those of the…

Consider a deterministic self-adjoint matrix X_n with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by…

The largest eigenvalue of the adjacency matrix of a network plays an important role in several network processes (e.g., synchronization of oscillators, percolation on directed networks, linear stability of equilibria of network coupled…

Given two real symmetric matrices, their eigenvalue configuration is the relative arrangement of their eigenvalues on the real line. In this paper, we consider the following problem: given two parametric real symmetric matrices and an…