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1. A standard Gaussian random matrix has full rank with probability 1 and is well-conditioned with a probability quite close to 1 and converging to 1 fast as the matrix deviates from square shape and becomes more rectangular. 2. If we…
The problem of membrane topology in the matrix model of M-theory is considered. The matrix regularization procedure, which makes a correspondence between finite-sized matrices and functions defined on a two-dimensional base space, is…
In this short note, we propose to extend differentiability (with respect to a multidimensional parameter) of a normalized eigenfunction associated to the simple, dominating eigenvalue of the weighted transfer operator for a uniformly…
The most general form of a marginal extended perturbation in a two-dimensional system is deduced from scaling considerations. It includes as particular cases extended perturbations decaying either from a surface, a line or a point for which…
The eigenvalue spacing of a uniformly chosen random finite unipotent matrix in its permutation action on lines is studied. We obtain bounds for the mean number of eigenvalues lying in a fixed arc of the unit circle and offer an approach…
We note that building a magnetic Laplacian from the Markov transition matrix, rather than the graph adjacency matrix, yields several benefits for the magnetic eigenmaps algorithm. The two largest benefits are that the embedding becomes more…
The classical Hermitian eigenvalue problem addresses the following question: What are the possible eigenvalues of the sum A+B of two Hermitian matrices A and B, provided we fix the eigenvalues of A and B. A systematic study of this problem…
The paper is briefly dealing with greater or lesser misused normalization in self-modeling/multivariate curve resolution (S/MCR) practice. The importance of the correct use of the ode solvers and apt kinetic illustrations are elucidated.…
Recently, three numerical methods for the computation of eigenvalues of singular matrix pencils, based on a rank-completing perturbation, a rank-projection, or an augmentation were developed. We show that all three approaches can be…
This paper deals with the problem of parameter estimation based on certain eigenspaces of the empirical covariance matrix of an observed multidimensional time series, in the case where the time series dimension and the observation window…
We investigate almost-degenerate perturbation theory of eigenvalue problems, using spectral projectors, also named density matrices. When several eigenvalues are close to each other, the coefficients of the perturbative series become…
We work with the Friedrichs extension of a one dimensional Schrodinger whose potential has a certain type of regular singularity near one end point. We study the effect on the eigenvalues of shrinking the region slightly near the end point.…
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…
We consider the Gaussian ensembles of random matrices and describe the normal modes of the eigenvalue spectrum, i.e., the correlated fluctuations of eigenvalues about their most probable values. The associated normal mode spectrum is…
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 eigenvalue statistics of a pair $(M_1,M_2)$ of $n\times n$ Hermitian matrices taken random with respect to the measure $$\frac{1}{Z_n}\exp\big(-n\Tr (V(M_1)+W(M_2)-\tau M_1M_2)\big) {\rm d}M_1 {\rm d} M_2 $$ can be described in terms of…
A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory results for linearizations need to be related back to matrix polynomials. In this paper we…
In this paper, new conditions for the stability of V-geometrically ergodic Markov chains are introduced. The results are based on an extension of the standard perturbation theory formulated by Keller and Liverani. The continuity and higher…
We study the convergence properties of a pair of learning algorithms (learning with and without memory). This leads us to study the dominant eigenvalue of a class of random matrices. This turns out to be related to the roots of the…
One-rank perturbations of Wigner matrices have been closely studied: let $P=\frac{1}{\sqrt{n}}A+\theta vv^T$ with $A=(a_{ij})_{1 \leq i,j \leq n} \in \mathbb{R}^{n \times n}$ symmetric, $(a_{ij})_{1 \leq i \leq j \leq n}$ i.i.d. with…