Since it is required that a projection be self-adjoint in the chosen Note that (1.1) depends on the scalar product Of the double canonical angles between I and I in the scalar product Same dimension, we are interested in providing estimates for the size of Non-singular matrices and their spectral subspaces I and are of the Given definite Hermitian matrix pairs (H, M) and, where H,, M, and are and such that H + M is a positive definite Ī matrix pair (H, M) is called definitizable if there exist scalars In a matrix-dependent scalar product, see. For a discussion of the geometry of an Euclidean space Our main contribution is to establish such an estimate for theĭefinitizable generalized eigenvalue problem in a matrix-dependent Alternatively, we can obtain estimates which feature onlyĪ posteriori distances (those in the spectrum of H) by reversing the Priori since only the separation d between the wanted and unwantedĬomponents of the spectrum of the unperturbed matrix H is appearing in In the case d = 0, we take 1/d = and theīound trivially holds. A generic estimate can be formulated as the following sinĢ-bound, which is taken from a recent paper by Albeverio and ![]() We use Ran() to denote the range ofĪ matrix. We freely associate the angle operator withĮither the subspaces or the corresponding projections whatever isĪppropriate in a given context. Introduce the notation for the associated subspaces and [MATHEMATICAL EXPRESSION NOT Since orthogonal projections are Hermitian (self-adjoint) and We useĮ() and toĭenote the spectral projection of H and E() associated to the REPRODUCIBLE IN ASCII] hold for the spectra of H and. Us consider matrices H and, and let the claims, and [MATHEMATICAL EXPRESSION NOT What follows, we use Spec(H) to denote the spectrum of a matrix H. Subspaces of the unperturbed and perturbed operators, respectively. Trigonometric function of the angle operator associated with spectral The main objective in these studies was to obtain a bound of a In particular, it influenced theĭevelopment of mathematical software for highly accurate solutions of Operator theory (scattering theory in mathematical physics) as well as Paper of Davis and Kahan, which had both fundamental importance in A sequence of three papers culminated with the cornerstone Under the influence of a perturbation see, e.g., and the references Davis on the rotation of invariant subspaces See for results in operator theory and for recent results in the context of matrix analysis.Īmong the most prominent results in this field of research is the One of the fundamental problems in operator theory or matrix analysis Invariant subspace of a matrix or operator A under a perturbation V is Controlling the size of the rotation of an matrix pairs, perturbation of eigenvectors, sin 2ĪMS subject classifications. The quality of the new boundsĪre illustrated in the numerical examples. Is measured in the matrix-dependent scalar product, and the boundsīelong to relative perturbation theory. H is a non-singular matrix which can be factorized as H = GJG*, J =ĭiag(1), and M is non-singular. The rotation of the eigenspaces of Hermitian matrix pairs (H, M), where In this paper we present new double angle theorems for ![]() APA style: Double angle theorems for definite matrix pairs. ![]() Double angle theorems for definite matrix pairs." Retrieved from 2016 Institute of Computational Mathematics 22 Sep. MLA style: "Double angle theorems for definite matrix pairs." The Free Library.
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