Let A be an n × n matrix with real eigenvalues λ1 ? … ? λn, and let 1 ? k < l ? n. Bounds involving trA and trA2 are introduced for λk/λl, (λk ? λl)/(λk + λl), and {kλk + (n ? l + 1)λl}2/{kλ2k + (n ? l + 1)λ2l}. Also included are conditions for λl >; 0 and for λk + λl > 0. 相似文献
Given a set of 2n real numbers λ1<λ2<?<λ2n, the authors describe the set {S} of n × n tridiagonal matrices with the property that each S can be completed to a 2n×2n tridiagonal matrix L with spec(L)={λ1, λ2,…,λ2n}. 相似文献
The inverse of a graph with the spectrum λ1, λ1, …λn (λ1≠0) is a graph with the spectrum 1/λ1,1/λ2,…,1/λn,.We present a purely graph-theoretic construction of the inverse ol a tree with a perfect matening. We apply this method for deriving results concerning the least nonnegative eigenvalue of a tree (called the dual index of a tree), including the best possible upper bound for the dual index of a tree in terms of a the number of its vertices. 相似文献
Let T(λ, ε ) = λ2 + λC + λεD + K be a perturbed quadratic matrix polynomial, where C, D, and K are n × n hermitian matrices. Let λ0 be an eigenvalue of the unperturbed matrix polynomial T(λ, 0). With the falling part of the Newton diagram of det T(λ, ε), we find the number of differentiable eigenvalues. Some results are extended to the general case L(λ, ε) = λ2 + λD(ε) + K, where D(ε) is an analytic hermitian matrix function. We show that if K is negative definite on Ker L(λ0, 0), then every eigenvalue λ(ε) of L(λ, ε) near λ0 is analytic. 相似文献
We show that a graph G on n ? q + 1 vertices (where q ? 2) has the chromatic polynomial P(G;λ) = λ(λ ? 1) … (λ ? q + 2) (λ ? q + 1)2 (λ ? q)n?q?1 if and only if G can be obtained from a q-tree Ton n vertices by deleting an edge contained in exactly q ? 1 triangles of T. Furthermore, we prove that these graphs are triangulated. 相似文献
Let M be a closed Riemannian manifold of dimension n. Let ?λ be an eigenfunction of the Laplace–Beltrami operator corresponding to an eigenvalue λ. We show that the volume of {?λ > 0} ∩ B is ≥C|B|/λn, where B is any ball centered at a point of the nodal set. We apply this result to prove that each nodal domain contains a ball of radius ≥C/λn. The results in this paper extend previous results of Nazarov, Polterovich, Sodin and of the author. 相似文献
A Jordan partition λ(m, n, p) = (λ1, λ2, … , λm) is a partition of mn associated with the expression of a tensor Vm ? Vn of indecomposable KG-modules into a sum of indecomposables, where K is a field of characteristic p and G a cyclic group of p-power order. It is standard if λi = m + n ? 2i + 1 for all i. We answer a recent question of Glasby, Praeger, and Xia who asked for necessary and sufficient conditions for λ(m, n, p) to be standard. 相似文献
We show that for every set Λ={λ1,λ2,…,λn} of real numbers such that , there exists a doubly stochastic matrix with spectrum Λ. We present an explicit construction of such a matrix. 相似文献
Let A be an n×n complex-valued matrix, all of whose principal minors are distinct from zero. Then there exists a complex diagonal matrix D, such that the spectrum of AD is a given set σ = {λ1,…,λn} in C. The number of different matrices D is at most n!. 相似文献
The (exterior) oblique derivative problem of potential theory is considered where l is a C(1,λ)-vector field on a regular boundary ?Ge in Euclidean space R 3 and the direction l at any point of ?Ge forms with the outside normal n an angle ? (l, n) satisfying cos ? (l, n) ≥ c > 0. An approximation of the uniquely determined solution is given by use of trial functions {Φn}n=0,1,… harmonic in some region containing Ge and suitable for numerical purpose (for instance: solid spherical harmonics, certain sequences of fundamental solutions). The system {l▽Φn+hΦn}n=0,1,… defined on ?Ge is shown to be closed and complete in the pre-Hilbert space C(0,λ)(?Ge) of λ-Hölder continuous functions. 相似文献