相关论文: Symplectic eigenvalues of positive-semidefinite ma…

In this paper, we introduce a particular class of matrices. We study the concept of a matrix to be \emph{balanced}. We study some properties of this concept in the context of matrix operations. We examine the behaviour of various matrix…

We consider the problem of writing an arbitrary symmetric matrix as the difference of two positive semidefinite matrices. We start with simple ideas such as eigenvalue decomposition. Then, we develop a simple adaptation of the Cholesky that…

The properties of the first (largest) eigenvalue and its eigenvector (first eigenvector) are investigated for large sparse random symmetric matrices that are characterized by bimodal degree distributions. In principle, one should be able to…

We prove the Central Limit Theorem for finite-dimensional vectors of linear eigenvalue statistics of submatrices of Wigner random matrices under the assumption that test functions are sufficiently smooth. We connect the asymptotic…

We investigate the spectral distribution of random matrix ensembles with correlated entries. We consider symmetric matrices with real valued entries and stochastically independent diagonals. Along the diagonals the entries may be…

The spectra of signed matrices have played a fundamental role in social sciences, graph theory, and control theory. In this work, we investigate the computational problems of identifying symmetric signings of matrices with natural spectral…

Over the past few years, trace regression models have received considerable attention in the context of matrix completion, quantum state tomography, and compressed sensing. Estimation of the underlying matrix from regularization-based…

This note contains a short proof of a classical result: any rational symplectic matrix can be put in diagonal form after right and left multiplication by integral symplectic matrices.

It is well known that the sum of negative (positive) eigenvalues of some finite Hermitian matrix $V$ is concave (convex) with respect to $V$. Using the theory of the spectral shift function we generalize this property to self-adjoint…

A method of computation of its terms is presented together with some stabilization results. As an application a characterization of symplectic harmonic manifolds is given and a relationship with the C-spectral sequence is indicated.

We consider the characterizations of positive definite as well as nonnegative definite quadratic forms in terms of the principal minors of the associated symmetric matrix. We briefly review some of the known proofs, including a classical…

Positive semidefinite matrices partitioned into a small number of Hermitian blocks have a remarkable property. Such a matrix may be written in a simple way from the sum of its diagonal blocks

A characterization of the general linear equation in standard form admitting a maximal symmetry algebra is obtained in terms of a simple set of conditions relating the coefficients of the equation. As a consequence, it is shown that in its…

We characterize the relationship between the singular values of a complex Hermitian (resp., real symmetric, complex symmetric) matrix and the singular values of its off-diagonal block. We also characterize the eigenvalues of an Hermitian…

In this paper, we introduce symmetric diagram matrices $A_{s+r,s}$ of size ${_{(s+r)}}C_s$ whose entries are $\{x_i\}_{min\{s,r\}}$. We compute the eigenvalues of symmetric diagram matrices using elementary row and column operations…

We study M-tensors and various properties of M-tensors are given. Specially, we show that the smallest real eigenvalue of M-tensor is positive corresponding to a nonnegative eigenvector. We propose an algorithm to find the smallest positive…

An approach, based on the Smith Normal Form, is introduced to study the spectra of symmetric matrices with a given graph. The approach serves well to explain how the path cover number (resp. diameter of a tree T) is related to the maximum…

Eigenvalue spectrum of the Laplacian on a metric graph with arbitrary but fixed vertex conditions is investigated in the limit as the lengths of all edges decrease to zero at the same rate. It is proved that there are exactly four possible…

We present a formula for the trace of any symmetric power of a $n\times n$ matrix (with coefficients in a field) in terms of the ordinary powers of the matrix, an arbitrarily chosen linear function which vanishes on the identity matrix, and…

We study ill-conditioned positive definite matrices that are disturbed by the sum of $m$ rank-one matrices of a specific form. We provide estimates for the eigenvalues and eigenvectors. When the condition number of the initial matrix tends…