Related papers: Small Eigenvalues of Large Hankel Matrices

We study linear eigenvalue statistics of band Hankel matrices with Brownian motion entries. We prove that, the centred, normalized linear eigenvalue statistics of band Hankel matrices obey a central limit theorem (CLT) type result. We also…

For an $n\times n$ Laplacian random matrix $L$ with Gaussian entries it is proven that the fluctuations of the largest eigenvalue and the largest diagonal entry of $L/\sqrt{n-1}$ are Gumbel. We first establish suitable non-asymptotic…

Let $G$ be a connected non-odd-bipartite hypergraph with even uniformity. The least H-eigenvalue of the signless Laplacian tensor of $G$ is simply called the least eigenvalue of $G$ and the corresponding H-eigenvectors are called the first…

Let $$ A=\left(\begin{array}{cc} H_1 & E^*\\ E & H_2\end{array}\right) \quad \hbox{ and } \quad \wtd A=\left(\begin{array}{cc} H_1 & O\\ O & H_2\end{array}\right)$$ be two $N$-by-$N$ Hermitian matrices with eigenvalues $\lambda_1 \ge \cdots…

We propose two different strategies to find eigenvalues and eigenvectors of a given, not necessarily Hermitian, matrix $A$. Our methods apply also to the case of complex eigenvalues, making the strategies interesting for applications to…

Hankel matrices are an important class of highly-structured matrices, arising across computational mathematics, engineering, and theoretical computer science. It is well-known that positive semidefinite (PSD) Hankel matrices are always…

We consider the smallest eigenvalues of perturbed Hermitian operators with zero modes, either topological or system specific. To leading order for small generic perturbation we show that the corresponding eigenvalues broaden to a Gaussian…

Suppose $X$ is an $N \times n$ complex matrix whose entries are centered, independent, and identically distributed random variables with variance $1/n$ and whose fourth moment is of order ${\mathcal O}(n^{-2})$. In the first part of the…

Let $X$ be a $p\times n$ independent identically distributed real Gaussian matrix with positive mean $\mu $ and variance $\sigma^2$ entries. The goal of this paper is to investigate the largest eigenvalue of the noncentral sample covariance…

We study the Hankel determinant generated by a deformed Hermite weight with one jump $w(z,t,\gamma)=e^{-z^2+tz}|z-t|^{\gamma}(A+B\theta(z-t))$, where $A\geq 0$, $A+B\geq 0$, $t\in\textbf{R}$, $\gamma>-1$ and $z\in\textbf{R}$. By using the…

In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from $n$ independent observations of a $p$-dimensional time series with iid components converge almost surely to $(1+\sqrt{\gamma})^2$…

Unlike an irreducible $Z$-matrices, a weakly irreducible $Z$-tensor $\mathcal{A}$ can have more than one eigenvector associated with the least H-eigenvalue. We show that there are finitely many eigenvectors of $\mathcal{A}$ associated with…

Inspired by the quantum computing algorithms for Linear Algebra problems [HHL,TaShma] we study how the simulation on a classical computer of this type of "Phase Estimation algorithms" performs when we apply it to solve the Eigen-Problem of…

We study the eigenvalues of the covariance matrix $\frac{1}{n}M^*M$ of a large rectangular matrix $M=M_{n,p}=(\zeta_{ij})_{1\leq i\leq p;1\leq j\leq n}$ whose entries are i.i.d. random variables of mean zero, variance one, and having finite…

In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight $w(x)=x^\alpha e^{-x-t/x}$, $x\in (0, \infty)$, $t>0$ and $\alpha>0$. When the matrix size $n\to\infty$, we obtain an asymptotic…

We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary and symplectic ensembles. In particular, we show that the…

In this paper we prove that the smallest eigenvalue $\lambda_1$ of the eigenvalue problem for a quasilinear elliptic systems introduced by de Th\'elin in \cite{DT}, is not only simple (in a suitable sense), but also isolated. Moreover, we…

Random Hermitian matrices are used to model complex systems without time-reversal invariance. Adding an external source to the model can have the effect of shifting some of the matrix eigenvalues, which corresponds to shifting some of the…

The smallest nonzero eigenvalue of the normalized Laplacian matrix of a graph has been extensively studied and shown to have many connections to properties of the graph. We here study a generalization of this eigenvalue, denoted $\lambda(G,…

Let $W_n= \frac{1}{\sqrt n} M_n$ be a Wigner matrix whose entries have vanishing third moment, normalized so that the spectrum is concentrated in the interval $[-2,2]$. We prove a concentration bound for $N_I = N_I(W_n)$, the number of…