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We prove some inequalities involving fourth central moment of a random variable that takes values in a given finite interval. Both discrete and continuous cases are considered. Bounds for the spread are obtained when a given $$n\times n$$ complex matrix has real eigenvalues. Likewise, we discuss bounds for the spans of polynomial equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the eigenvalue variation for any perturbation of Hermitian matrices, and we obtain two perturbation bounds. The first bound always improves the existing bound, and the second bound also improves the existing one under a suitable condition. A simple example is given for comparing these bounds. [ABSTRACT FROM AUTHOR]

Using the method of seeds and branch duplication, it is shown that for every tree of diameter, there is an Hermitian matrix with as few asdistinct eigenvalues (a known lower bound). For diameter 7, some trees require 8 distinct eigenvalues, but no more; the seeds for which 7 and 8 are the worst case are classified. For trees of diameter, it is shown that, in general, the minimum number of distinct eigenvalues is bounded by a function of. Many trees of high diameter permit as few of distinct eigenvalues as the diameter and a conjecture is made that all linear trees are of this type. Several other specific, related observations are made. [ABSTRACT FROM PUBLISHER]

SUMMARY In this paper, we present some new interlacing properties about the bounds of the eigenvalues for rank-one modification of Hermitian matrix, whose eigenvalues are different and spectral decomposition also needs to be known. Numerical examples demonstrate the efficiency of the proposed method and support our theoretical results.Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]