The Cauchy double alternant and divided differences.
The classification of edges and the change in multiplicity of an eigenvalue of a real symmetric matrix resulting from the change in an edge value
We take as given a real symmetric matrix A, whose graph is a tree T, and the eigenvalues of A, with their multiplicities. Each edge of T may then be classified in one of four categories, based upon the change in multiplicity of a particular eigenvalue, when the edge is removed (i.e. the corresponding entry of A is replaced by 0).We show a necessary and suficient condition for each possible classification of an edge. A special relationship is observed among 2-Parter edges, Parter edges and singly...
The combinatorial structure of eventually nonnegative matrices.
The comparison of spectrum of normalizable matrices
The convex analysis of unitarily invariant matrix functions.
The Direct and Inverse Spectral Problems for some Banded Matrices
2000 Mathematics Subject Classification: 15A29.In this paper we introduced a notion of the generalized spectral function for a matrix J = (gk,l)k,l = 0 Ґ, gk,l О C, such that gk,l = 0, if |k-l | > N; gk,k+N = 1, and gk,k-N № 0. Here N is a fixed positive integer. The direct and inverse spectral problems for such matrices are stated and solved. An integral representation for the generalized spectral function is obtained.
The distribution of eigenvalues of randomized permutation matrices
In this article we study in detail a family of random matrix ensembles which are obtained from random permutations matrices (chosen at random according to the Ewens measure of parameter ) by replacing the entries equal to one by more general non-vanishing complex random variables. For these ensembles, in contrast with more classical models as the Gaussian Unitary Ensemble, or the Circular Unitary Ensemble, the eigenvalues can be very explicitly computed by using the cycle structure of the permutations....
The eigenvalue derivatives of linear damped systems
The eigenvalue distribution on Schur complement of nonstrictly diagonally dominant matrices and general -matrices.
The Eigenvalues and Singular Values of Matrix Sums and Product, VIII: Displacement of Indices
The eigenvalues of Hermite and rational spectral differentiation matrices.
The empirical distribution of the eigenvalues of a Gram matrix with a given variance profile
The fan graph is determined by its signless Laplacian spectrum
Given a graph , if there is no nonisomorphic graph such that and have the same signless Laplacian spectra, then we say that is -DS. In this paper we show that every fan graph is -DS, where and .
The higher rank numerical range of nonnegative matrices
In this article the rank-k numerical range ∧k (A) of an entrywise nonnegative matrix A is investigated. Extending the notion of elements of maximum modulus in ∧k (A), we examine their location on the complex plane. Further, an application of this theory to ∧k (L(λ)) of a Perron polynomial L(λ) is elaborated via its companion matrix C L.
The importance of each node in a structure of links: Google-PageRank.
The inertia set of nonnegative symmetric sign pattern with zero diagonal
The inertia set of a symmetric sign pattern is the set , where denotes the inertia of real symmetric matrix , and denotes the sign pattern class of . In this paper, a complete characterization on the inertia set of the nonnegative symmetric sign pattern in which each diagonal entry is zero and all off-diagonal entries are positive is obtained. Further, we also consider the bound for the numbers of nonzero entries in the nonnegative symmetric sign patterns with zero diagonal that require...
The inverse eigenvalue and inertia problems for minimum rank two graphs.
The Jordan forms of and .
The Laplacian spectrum of some digraphs obtained from the wheel
The problem of distinguishing, in terms of graph topology, digraphs with real and partially non-real Laplacian spectra is important for applications. Motivated by the question posed in [R. Agaev, P. Chebotarev, Which digraphs with rings structure are essentially cyclic?, Adv. in Appl. Math. 45 (2010), 232-251], in this paper we completely list the Laplacian eigenvalues of some digraphs obtained from the wheel digraph by deleting some arcs.
The Laplacian spread of graphs
The Laplacian spread of a graph is defined as the difference between the largest and second smallest eigenvalues of the Laplacian matrix of the graph. In this paper, bounds are obtained for the Laplacian spread of graphs. By the Laplacian spread, several upper bounds of the Nordhaus-Gaddum type of Laplacian eigenvalues are improved. Some operations on Laplacian spread are presented. Connected -cyclic graphs with vertices and Laplacian spread are discussed.