9 research outputs found
Bounds on changes in Ritz values for a perturbed invariant subspace of a Hermitian matrix
The Rayleigh-Ritz method is widely used for eigenvalue approximation. Given a
matrix with columns that form an orthonormal basis for a subspace \X, and
a Hermitian matrix , the eigenvalues of are called Ritz values of
with respect to \X. If the subspace \X is -invariant then the Ritz
values are some of the eigenvalues of . If the -invariant subspace \X
is perturbed to give rise to another subspace \Y, then the vector of absolute
values of changes in Ritz values of represents the absolute eigenvalue
approximation error using \Y. We bound the error in terms of principal angles
between \X and \Y. We capitalize on ideas from a recent paper [DOI:
10.1137/060649070] by A. Knyazev and M. Argentati, where the vector of absolute
values of differences between Ritz values for subspaces \X and \Y was
weakly (sub-)majorized by a constant times the sine of the vector of principal
angles between \X and \Y, the constant being the spread of the spectrum of
. In that result no assumption was made on either subspace being
-invariant. It was conjectured there that if one of the trial subspaces is
-invariant then an analogous weak majorization bound should only involve
terms of the order of sine squared. Here we confirm this conjecture.
Specifically we prove that the absolute eigenvalue error is weakly majorized by
a constant times the sine squared of the vector of principal angles between the
subspaces \X and \Y, where the constant is proportional to the spread of
the spectrum of . For many practical cases we show that the proportionality
factor is simply one, and that this bound is sharp. For the general case we can
only prove the result with a slightly larger constant, which we believe is
artificial.Comment: 12 pages. Accepted to SIAM Journal on Matrix Analysis and
Applications (SIMAX
Rayleigh-Ritz majorization error bounds of the mixed type
The absolute change in the Rayleigh quotient (RQ) for a Hermitian matrix with
respect to vectors is bounded in terms of the norms of the residual vectors and
the angle between vectors in [\doi{10.1137/120884468}]. We substitute
multidimensional subspaces for the vectors and derive new bounds of absolute
changes of eigenvalues of the matrix RQ in terms of singular values of residual
matrices and principal angles between subspaces, using majorization. We show
how our results relate to bounds for eigenvalues after discarding off-diagonal
blocks or additive perturbations.Comment: 20 pages, 1 figure. Accepted to SIAM Journal on Matrix Analysis and
Application
Angles Between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods
We define angles from-to and between infinite dimensional subspaces of a
Hilbert space, inspired by the work of E. J. Hannan, 1961/1962 for general
canonical correlations of stochastic processes. The spectral theory of
selfadjoint operators is used to investigate the properties of the angles,
e.g., to establish connections between the angles corresponding to orthogonal
complements. The classical gaps and angles of Dixmier and Friedrichs are
characterized in terms of the angles. We introduce principal invariant
subspaces and prove that they are connected by an isometry that appears in the
polar decomposition of the product of corresponding orthogonal projectors.
Point angles are defined by analogy with the point operator spectrum. We bound
the Hausdorff distance between the sets of the squared cosines of the angles
corresponding to the original subspaces and their perturbations. We show that
the squared cosines of the angles from one subspace to another can be
interpreted as Ritz values in the Rayleigh-Ritz method, where the former
subspace serves as a trial subspace and the orthogonal projector of the latter
subspace serves as an operator in the Rayleigh-Ritz method. The Hausdorff
distance between the Ritz values, corresponding to different trial subspaces,
is shown to be bounded by a constant times the gap between the trial subspaces.
We prove a similar eigenvalue perturbation bound that involves the gap squared.
Finally, we consider the classical alternating projectors method and propose
its ultimate acceleration, using the conjugate gradient approach. The
corresponding convergence rate estimate is obtained in terms of the angles. We
illustrate a possible acceleration for the domain decomposition method with a
small overlap for the 1D diffusion equation.Comment: 22 pages. Accepted to Journal of Functional Analysi
Angles, Majorization, Wielandt Inequality and Applications
In this thesis we revisit two classical definitions of angle in an inner product space: real-part angle and Hermitian angle. Special attention is paid to Krein’s inequality and its
analogue. Some applications are given, leading to a simple proof of a basic lemma for a trace inequality of unitary matrices and also its extension. A brief survey on recent results of angles between subspaces is presented. This naturally brings us to the world of majorization. After introducing the notion of majorization, we present some classical as well as recent results on eigenvalue majorization. Several new norm inequalities are derived
by making use of a powerful decomposition lemma for positive semidefinite matrices. We also consider coneigenvalue majorization. Some discussion on the possible generalization of the majorization bounds for Ritz values is presented. We then turn to a basic notion in
convex analysis, the Legendre-Fenchel conjugate. The convexity of a function is important in finding the explicit expression of the transform for certain functions. A sufficient convexity condition is given for the product of positive definite quadratic forms. When the number of quadratic forms is two, the condition is also necessary. The condition is in terms of the condition number of the underlying matrices. The key lemma in our derivation is
found to have some connection with the generalized Wielandt inequality. A new inequality between angles in inner product spaces is formulated and proved. This leads directly to a concise statement and proof of the generalized Wielandt inequality, including a simple description of all cases of equality. As a consequence, several recent results in matrix analysis and inner product spaces are improved
La dispersión espectral y sus aplicaciones en el análisis matricial y la teoría de operadores
La teoría de perturbaciones constituye un tópico de estudio dentro del análisis matricial y la teoría de operadores. Asociados a este tópico, se encuentran los temas clásicos de álgebra lineal numérica y teoría de la aproximación.
En este contexto, podemos pensar en el estudio de la sensibilidad de los valores de Ritz y los cocientes de Rayleigh de matrices autoadjuntas, es decir, los cambios en los autovalores de compresiones de matrices autoadjuntas, que es un campo de investigación sólido y bien establecido en estas áreas.Facultad de Ciencias Exacta