Let G be a finite, connected, undirected graph, without loops or multiple edges. If v is a vertex of G, the degree of v, p {v), is the number of edges emanating from v. EL Brooks has shown [1] that &^ l+ maxp (v)(1) veG where к is the chromatic number of G, with eq...
We show that the Jackiw-Rebbi and Goldstone-Wilczek hamiltonians describing a two-component Dirac fermion in bounded background field and confined to a finite one-dimensional box has self-adjoint extensions which are labelled by a 2 × 2 unitary matrix U. The fractional part of the eigenvalue...
. . , λn be the eigenvalues of a matrix A, let x1, x2, . . . , xn be a set of corresponding eigenvectors, let Λ denote the n-by-n diagonal matrix with the λj on the diagonal, and let X denote the n-by-n matrix whose jth column is xj. Then AX = XΛ. It is ...
A matrix A \in M_{n}(\mathbb{C}) is said to be conjugate-normal if AA^{*} = \overline{A^{*}A}. Complex symmetric, skew-symmetric, and unitary matrices are special subclasses of conjugate-normal matrices. For the properties and characterizations of this kind of matrices, readers ar...
Krause, “Bounds for the variation of the roots of a polynomial and the eigenvalues of a matrix,” Linear Algebra and Its Applications, vol. 142, pp. 195–209, 1990. CrossRef N. Gozlan, “A characterization of dimension free concentration in terms of transportation inequalities,” The ...
partofA.ThematrixShasadistributioncommonlyknownastheGaussian UnitaryEnsemble;thismatrixdistributionandthejointdistributionofits(real) eigenvalueshavebeenwellstudied.Foragoodreferenceonthesubject,seeMehta [10]. Wedrawbelowhistogramsofnormalizedeigenvaluestakenfromthisdistribu- tion,theknowntheoreticaldistribution(see[10...
ZLAQR4 - ZLAQR4 compute the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**H, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors ...
is the order of the general matrixA. Specified as: an integer;n0. a is the general matrixAof ordern. Specified as: anldaby (at least)narray, containing numbers of the data type indicated inTable 1. lda is the leading dimension...
(t−).InthecommutativecasealltheeigenvaluesjumpattimetonlywhenthejumpofX isoffullrank.Inthenoncommutativecase,Xjumpsattimetifandonlyifalltheeigenvaluesjumpat thattimewhenthejumpofXisofrankone. Keywords:Infinitelydivisiblerandommatrix,Dyson–Brownianmotion,Bercovici–Patabijection, matrixsemimartingale,eigen...
A common way of finding the poles of a meromorphic function f in a domain, where an explicit expression of f is unknown but f can be evaluated at any given z, is to interpolate f by a rational function such that at prescribed sample points , and then find the roots of q. This is ...