We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l 2(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J...

We study block diagonalization of matrices induced by resolutions of the unit matrix into the sum of idempotent matrices. We show that the block diagonal matrices have disjoint spectra if and only if each idempotent matrix in the inducing resolution double commutes with the given matrix. Applications include a new characterization of an eigenprojection and of the Drazin inverse of a given matrix.

In this paper, we give a new bound for the largest singular value of nonnegative rectangular tensors when m = n, which is tighter than the bound provided by Yang and Yang in “Singular values of nonnegative rectangular tensors”, Front. Math. China, 2011, 6, 363-378.

Let $G$ be a simple connected graph of order $n$ with degree sequence $(d_1,d_2,...,d_n)$. Denote ${{(}^{\alpha }t)}_i={\sum }_{j:i\sim j}{d}_j^{\alpha }$, ${{(}^{\alpha }m)}_i={{(}^{\alpha }t)}_i/d_i^{\alpha }$ and ${{(}^{\alpha }N)}_i={\sum }_{j:i\sim j}{{(}^{\alpha }t)}_j$, where $\alpha$ is a real number. Denote by ${\lambda }_1\left(G\right)$ and ${\mu }_1\left(G\right)$ the spectral radius of the adjacency matrix and the Laplacian matrix of $G$, respectively. In this paper, we present some upper and lower bounds of ${\lambda }_1\left(G\right)$ and ${\mu }_1\left(G\right)$ in terms of ${{(}^{\alpha }t)}_i$, ${{(}^{\alpha }m)}_i$ and ${{(}^{\alpha }N)}_i$. Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results.

We are concerned with bounds of the matrix eigenvalues and its exponential. Combining the Lyapunov equation with the weighted logarithmic matrix norm technique, four sequences are presented to locate eigenvalues of a matrix. Based on the relations between the real parts of the eigenvalues and the weighted logarithmic matrix norms, we derive both lower and upper bounds of the matrix exponential, which complement and improve the existing results in the literature. Some numerical examples are also...

A total dominating set in a graph $G$ is a subset $X$ of $V\left(G\right)$ such that each vertex of $V\left(G\right)$ is adjacent to at least one vertex of $X$. The total domination number of $G$ is the minimum cardinality of a total dominating set. A function $f:V\left(G\right)\to \{-1,1\}$ is a signed dominating function (SDF) if the sum of its function values over any closed neighborhood is at least one. The weight of an SDF is the sum of its function values over all vertices. The signed domination number of $G$ is the minimum weight of an SDF on $G$. In this paper...

Let $A$ be an $n\times n$ symmetric, irreducible, and nonnegative matrix whose eigenvalues are ${\lambda }_1>{\lambda }_2\ge ...\ge {\lambda }_n$. In this paper we derive several lower and upper bounds, in particular on ${\lambda }_2$ and ${\lambda }_n$, but also, indirectly, on $\mu ={max}_{2\le i\le n}\left|{\lambda }_i\right|$. The bounds are in terms of the diagonal entries of the group generalized inverse, $Q^{\#}$, of the singular and irreducible M-matrix $Q={\lambda }_{1}I-A$. Our starting point is a spectral resolution for $Q^{\#}$. We consider the case of equality in some of these inequalities and we apply our results to the algebraic connectivity of undirected...

The coefficients of the greatest common divisor of two polynomials $f$ and $g$ (GCD$(f,g)$) can be obtained from the Sylvester subresultant matrix $S_j(f,g)$ transformed to lower triangular form, where $1\le j\le d$ and $d=$ deg(GCD$(f,g)$) needs to be computed. Firstly, it is supposed that the coefficients of polynomials are given exactly. Transformations of $S_j(f,g)$ for an arbitrary allowable $j$ are in details described and an algorithm for the calculation of the GCD$(f,g)$ is formulated. If inexact polynomials are given, then an approximate greatest...

We prove that any quaternionic matrix of order n ≤3 admits a characteristic function, whose roots are the left eigenvalues, that satisfes Cayley-Hamilton theorem.

In this paper, we study the fluctuations of the extreme eigenvalues of a spiked finite rank deformation of a Hermitian (resp. symmetric) Wigner matrix when these eigenvalues separate from the bulk. We exhibit quite general situations that will give rise to universality or non-universality of the fluctuations, according to the delocalization or localization of the eigenvectors of the perturbation. Dealing with the particular case of a spike with multiplicity one, we also establish a necessary and...