Results 1 
8 of
8
Bounds on changes in Ritz values for a perturbed invariant subspace of a Hermitian matrix
 SIAM J. Matrix Anal. Appl
"... Abstract. The RayleighRitz method is widely used for eigenvalue approximation. Given a matrix X with columns that form an orthonormal basis for a subspace X, and a Hermitian matrix A, the eigenvalues of X H AX are called Ritz values of A with respect to X. If the subspace X is Ainvariant then the ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
(Show Context)
Abstract. The RayleighRitz method is widely used for eigenvalue approximation. Given a matrix X with columns that form an orthonormal basis for a subspace X, and a Hermitian matrix A, the eigenvalues of X H AX are called Ritz values of A with respect to X. If the subspace X is Ainvariant then the Ritz values are some of the eigenvalues of A. If the Ainvariant subspace X is perturbed to give rise to another subspace Y, then the vector of absolute values of changes in Ritz values of A 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 A. In that result no assumption was made on either subspace being Ainvariant. It was conjectured there that if one of the trial subspaces is Ainvariant then an analogous weak majorization bound should be much stronger as it 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 A. 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. Key words. Hermitian matrices, angles between subspaces, majorization, Lidskii’s eigenvalue theorem, perturbation bounds, Ritz values, RayleighRitz method, invariant subspace.
MAJORIZATION BOUNDS FOR RITZ VALUES OF HERMITIAN MATRICES
, 2008
"... Given an approximate invariant subspace we discuss the effectiveness of majorization bounds for assessing the accuracy of the resulting RayleighRitz approximations to eigenvalues of Hermitian matrices. We derive a slightly stronger result than previously for the approximation of k extreme eigenvalu ..."
Abstract
 Add to MetaCart
(Show Context)
Given an approximate invariant subspace we discuss the effectiveness of majorization bounds for assessing the accuracy of the resulting RayleighRitz approximations to eigenvalues of Hermitian matrices. We derive a slightly stronger result than previously for the approximation of k extreme eigenvalues, and examine some advantages of these majorization bounds compared with classical bounds. From our results we conclude that the majorization approach appears to be advantageous, and that there is probably much more work to be carried out in this direction.
Angles Between Subspaces and Their Tangents
, 2013
"... Principal angles between subspaces (PABS) (also called canonical angles) serve as a classical tool in mathematics, statistics, and applications, e.g., data mining. Traditionally, PABS are introduced via their cosines. The cosines and sines of PABS are commonly defined using the singular value decomp ..."
Abstract
 Add to MetaCart
Principal angles between subspaces (PABS) (also called canonical angles) serve as a classical tool in mathematics, statistics, and applications, e.g., data mining. Traditionally, PABS are introduced via their cosines. The cosines and sines of PABS are commonly defined using the singular value decomposition. We utilize the same idea for the tangents, i.e., explicitly construct matrices, such that their singular values are equal to the tangents of PABS, using several approaches: orthonormal and nonorthonormal bases for subspaces, as well as projectors. Such a construction has applications, e.g., in analysis of convergence of subspace iterations for eigenvalue problems.
Maximization of the Sum of the Trace Ratio on the Stiefel Manifold
, 2013
"... We are concerned with the maximization of trpV J AV q trpV J BV q ` trpV J CV q over the Stiefel manifold tV P Rmˆℓ V JV “ Iℓu pℓ ă mq, where B is a given symmetric and positive definite matrix, A and C are symmetric matrices, and trp ¨ q is the trace of a square matrix. This is a subspace version ..."
Abstract
 Add to MetaCart
We are concerned with the maximization of trpV J AV q trpV J BV q ` trpV J CV q over the Stiefel manifold tV P Rmˆℓ V JV “ Iℓu pℓ ă mq, where B is a given symmetric and positive definite matrix, A and C are symmetric matrices, and trp ¨ q is the trace of a square matrix. This is a subspace version of the maximization problem studied in [Zhang, Comput. Optim. Appl., 54(2013), 111139], which arises from realworld applications in, for example, the downlink of a multiuser MIMO system and the sparse Fisher discriminant analysis in pattern recognition. We establish necessary conditions for both the local and global maximizers and connect the problem with a nonlinear extreme eigenvalue problem. The necessary condition for the global maximizers offers deep insights into the problem on the one hand and leads to a selfconsistentfield (SCF) iteration on the other hand. We analyze the global and local convergence of the SCF iteration, and show that the necessary condition for the global maximizers is fulfilled at any convergent point of the sequences of approximations generated by the SCF iteration. This is one of the advantages of the SCF iteration over optimizationbased methods. Preliminary numerical tests are reported and show that the SCF iteration is much more efficient than manifoldbased optimization methods.
OF SELFADJOINT OPERATORS ∗
, 2013
"... The absolute change in the Rayleigh quotient (RQ) is bounded in this paper in terms of the norm of the residual and the change in the vector. If x is an eigenvector of a selfadjoint bounded operator A in a Hilbert space, then the RQ of the vector x, denoted by (x), is an exact eigenvalue of A. In t ..."
Abstract
 Add to MetaCart
(Show Context)
The absolute change in the Rayleigh quotient (RQ) is bounded in this paper in terms of the norm of the residual and the change in the vector. If x is an eigenvector of a selfadjoint bounded operator A in a Hilbert space, then the RQ of the vector x, denoted by (x), is an exact eigenvalue of A. In this case, the absolute change of the RQ j(x) (y)j becomes the absolute error for an eigenvalue (x) of A approximated by the RQ (y) on a given vector y: There are three traditional kinds of bounds for eigenvalue errors: a priori bounds via the angle between vectors x and y; a posteriori bounds via the norm of the residual Ay (y)y of vector y; mixed type bounds using both the angle and the norm of the residual. We propose a unifying approach to prove known bounds of the spectrum, analyze their sharpness, and derive new sharper bounds. The proof approach is based on novel RQ vector perturbation identities.
Angles Between Infinite Dimensional Subspaces with Applications to the RayleighRitz and Alternating Projectors Methods ✩
"... We define angles fromto and between infinite dimensional subspaces of a Hilbert space using the spectra of the product of corresponding orthogonal projectors. Our definition is inspired by the work of E. J. Hannan, 1961/1962, who had suggested such an approach for general canonical correlations of ..."
Abstract
 Add to MetaCart
We define angles fromto and between infinite dimensional subspaces of a Hilbert space using the spectra of the product of corresponding orthogonal projectors. Our definition is inspired by the work of E. J. Hannan, 1961/1962, who had suggested such an approach for general canonical correlations of stochastic processes. Subsequent later developments for the infinite dimensional case, however, have been mostly limited to the gap and the minimum gap between subspaces despite the active work concerning all the angles in finite dimensional spaces. Our paper is intended to revive the interest in the angles between infinite dimensional subspaces and provides a foundation for work in applications, e.g., on canonical correlations for functional data. We use the spectral theory of selfadjoint operators to investigate the properties A preliminary version is available at
JOURNAL? (????), 1–15 DOI 10.1515/JOURNAL??????????? Angles
"... between subspaces and their tangents † ..."