Related papers: Eigenvalues of real symmetric matrices
We consider real non-symmetric matrices and their factorisation as a product of real symmetric matrices. The number of complex eigenvalues of the original matrix reveals restrictions on such factorisations as we shall prove.
In this paper we express the eigenvalues of a sort of real heptadiagonal symmetric matrices as the zeros of explicit rational functions establishing upper and lower bounds for each of them. From these prescribed eigenvalues we compute also…
In this short note, we present a novel method for computing exact lower and upper bounds of eigenvalues of a symmetric tridiagonal interval matrix. Compared to the known methods, our approach is fast, simple to present and to implement, and…
We study the real algebraic variety of real symmetric matrices with eigenvalue multiplicities determined by a partition. We present formulas for the dimension and Euclidean distance degree. We give a parametrization by rational functions.…
In this paper, we give estimates for both upper and lower bounds of eigenvalues of a simple matrix. The estimates are shaper than the known results.
Given an integral domain A we consider the set of all integral elements over A that can occur as an eigenvalue of a symmetric matrix over A. We give a sufficient criterion for being such an element. In the case where A is the ring of…
The structured pseudospectra of a matrix A are sets of complex numbers that are eigenvalues of matrices X which are near to A and have the same entries as A at a fixed set of places. The sum of multiplicities of the eigenvalues of X inside…
We establish new bounds on the minimum number of distinct eigenvalues among real symmetric matrices with nonzero off-diagonal pattern described by the edges of a graph and apply these to determine the minimum number of distinct eigenvalues…
We investigate convexity properties of the set of eigenvalue tuples of $n\times n$ real symmetric matrices, whose all $k\times k$ (where $k\leq n$ is fixed) minors are positive semidefinite. It is proven that the set…
We prove that few largest (and most important) eigenvalues of random symmetric matrices of various kinds are very strongly concentrated. This strong concentration enables us to compute the means of these eigenvalues with high precision. Our…
We develop several methods, based on the geometric relationship between the eigenspaces of a matrix and its adjoint, for determining whether a square matrix having distinct eigenvalues is unitarily equivalent to a complex symmetric matrix.…
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…
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…
We show that a rank-three symmetric matrix with exactly one negative eigenvalue can have arbitrarily large nonnegative rank.
We derive a sharp inequality relating the second and fourth elementary symmetric functions of the eigenvalues of a trace-free matrix and give two applications. First, we give a new proof of the classification of conformally flat…
We consider the eigenvalue problem for the case where the input matrix is symmetric and its entries perturb in some given intervals. We present a characterization of some of the exact boundary points, which allows us to introduce an inner…
This paper establishes new upper bounds for the sum of the $k$ largest eigenvalues of symmetric matrices. When applied to the adjacency matrix of a graph, our results improve upon a related bound due to Mohar {\bf [On the sum of k largest…
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…
The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product.
This paper establishes a comparison theorem for the maximum eigenvalue of a sum of independent random symmetric matrices. The theorem states that the maximum eigenvalue of the matrix sum is dominated by the maximum eigenvalue of a Gaussian…