Characterizations and lower bounds for the spread of a normal matrix (original) (raw)

Linear Algebra and its Applications, 1985

A characterization of the spread of a normal matrix is used to derive several simple lower bounds for the spread. Comparisons are then made with several known bounds.

Note on the spread of real symmetric matrices with entries in fixed interval

Linear Algebra and its Applications, 2021

The spread of a matrix is defined as the maximum of distances between any two eigenvalues of that matrix. In this paper we investigate spread maximization as a function on compact convex subset of the set of real symmetric matrices. We provide some general results and further, we study spread maximizing problem on Sn [a, b](the set of symmetric matrices with entries restricted to the interval [a, b]). In particular, we develop some results by X. Zhan (see [13]), S. M. Fallat and J. J. Xing (see [3]).

On the Distance Between Normal Matrices

Proceedings of the American Mathematical Society, 1990

The least upper bound for the norm distance between two normal matrices is given in terms of their eigenvalues exclusively, thus solving a problem which appears to be long open.

On star-centers of some generalized numerical ranges and diagonals of normal matrices

Linear Algebra and its Applications, 2001

For any n × n matrices A and C, we consider the star-centers of three sets, namely, the C-numerical range W C (A) of A, the set diag U(A) of diagonals of matrices in the unitary orbit of A, and the set S(A) of matrices whose C-numerical ranges are contained in W C (A) for all C. For normal matrices A, we show that the set of star-centers of W A * (A) is a bounded closed real interval, and give complete description of the sets of star-centers of diag U(A) and of S(A). In particular, we show that if A is normal with noncollinear eigenvalues, then each of S(A) and diag U(A) has exactly one star-center. For general square matrices A, we also give sufficient conditions for the sets of star-centers of diag U(A) and of S(A) to be singleton sets.

On spectral spread of generalized distance matrix of a graph

Linear and Multilinear Algebra, 2020

For a simple connected graph G, let D(G), T r(G), D L (G) and D Q (G), respectively be the distance matrix, the diagonal matrix of the vertex transmissions, distance Laplacian matrix and the distance signless Laplacian matrix of a graph G. The convex linear combinations D α (G) of T r(G) and D(G) is defined as D α (G) = αT r(G) + (1 − α)D(G), 0 ≤ α ≤ 1. As D 0 (G) = D(G), 2D 1 2 (G) = D Q (G), D 1 (G) = T r(G) and D α (G) − D β (G) = (α − β)D L (G), this matrix reduces to merging the distance spectral, distance Laplacian spectral and distance signless Laplacian spectral theories. Let ∂ 1 (G) ≥ ∂ 2 (G) ≥ • • • ≥ ∂ n (G) be the eigenvalues of D α (G) and let D α S(G) = ∂ 1 (G) −∂ n (G) be the generalized distance spectral spread of the graph G. In this paper, we obtain some bounds for the generalized distance spectral spread D α (G). We also obtain relation between the generalized distance spectral spread D α (G) and the distance spectral spread S D (G). Further, we obtain the lower bounds for D α S(G) of bipartite graphs involving different graph parameters and we characterize the extremal graphs for some cases. We also obtain lower bounds for D α S(G) in terms of clique number and independence number of the graph G and characterize the extremal graphs for some cases.

Bounds on the -spread of a graph

Linear Algebra and its Applications, 2010

The spread s(M ) of an n × n complex matrix M is s(M ) = max ij |λ i − λ j |, where the maximum is taken over all pairs of eigenvalues of M , λ i , 1 ≤ i ≤ n, [9] and . Based on this concept, Gregory et al.

Bounds on distances between eigenvalues

Linear Algebra and its Applications, 1984

Explicit (computable) lower and upper bounds on the distances between a given real eigenvalue of a real square matrix and the remaining (not necessarily real) eigenvalues of the matrix are developed.