Jane M. Day

SINGULAR VALUE INEQUALITY AND GRAPH ENERGY CHANGE

The energy of a graph is the sum of the singular values of its adjacency matrix. A classic inequality for singular values of a matrixsum, including its equality case, is used to study how the energy of a graph changes when edges are removed. One sharp bound and one bound that is never sharp, for the change in graph energy when the edges of a nonsingular induced subgraph are removed, are established. A graph is nonsingular if its adjacency matrixis nonsingular. 1. Singular value inequality for matrix sum. Let X be an n × n complex matrix and denote its singular values by s1(X) ≥ s2(X) ≥ · ·· ≥sn(X) ≥ 0. If X has real eigenvalues only, denote its eigenvalues by λ1(X) ≥ λ2(X) ≥ · ·· ≥λn(X). Define |X| = √ XX ∗ which is positive semi-definite, and note that λi(|X| )= si(X) for all i .W e w riteX ≥ 0t o meanX is positive semi-definite. We are interested in the following singular value inequality for a matrix sum: n � i=1 si(A + B) ≤ n � i=1 si(A )+ n � i=1 si(B)

The spectrum of a Hermitian matrix sum

Abstract An announcement by Lidskii, (B.V. Lidskii, Functional Anal. Appl. 10 (1982) 76–77 (Russian), 139–140 (English)), claimed to establish the explicit description of the spectrum of a Hermitian matrix sum in the form conjectured by Horn (A. Horn, Pacific J. Math. 16 (1962) 225–241), but no supporting proof has been published. This paper begins an analysis of the claim, and is the first step towards bridging the distance between the elementary methods of Horns (1962) paper and the partial solution of the same problem using noncommutative harmonic analysis by Dooley et al. (I. Dooley, R. Repka, N. Wildberger, Lin. Mult. Alg. 36 (1993) 79–102). The methods of Horn/Lidskii lead to combinatorial issues of independent interest.

Clan acts and codimension

Let (T, X) be a continuum act, let cd X=n and suppose A is a T-ideal (i.e., a T-invariant subspace of X), such that Hn(A)≠0. We prove that A is a minimal T-ideal iff A=Gx for some x∈X and maximal group G in the minimal ideal of T. Moreover, if these conditions are satisfied, then A is the only minimal T-ideal and also is the unique floor for every nonzero element of Hn(X). We need and also prove here an improved version of the Tube Theorem [3], and this corollary: if (G, X) is an intransitive transformation group with G compact, X locally compact and finite dimensional, and X/G connected, then dimension Gx

Some properties of the campbell baker hausdorff series

Some convergence properties for the Campbell-Baker-Hausdorff series for the logarithm of a product of exponentials are established.

Preaffine semigroups and convex matrix semigroups

Algebraic conditions are given which guarantee that a semigroup on a subset of real topological vector space can be embedded in a convex matrix semigroup. We also study when the minimal ideal of a convex matrix semigroup will be convex.