Localization in Matrix Computations: Theory and Applications

Detecting localized eigenstates of linear operators

“…In some situations, the localization of the eigenvectors follows from the gap assumption alone, for instance when the matrix A comes from a local discretization of a differential operator, as established in [2,13]. The theorem can be also extended to the case that the spectral projector to the low-lying eigenspace is localized, while individual eigenfunctions are not necessarily localized, in particular when the eigenvalues are near degenerate (see, e.g., [3]). …”

Bounds for the decay of the entries in inverses and Cauchy–Stieltjes functions of certain sparse, normal matrices

“…Off‐diagonal decay behavior in functions of banded matrices has long been studied, beginning with special emphasis on the matrix inverse and the matrix exponential . Further results on other classes of functions can be found in previous works; see also the recent survey by Benzi …”

“…This class of matrices includes (shifted) skew‐Hermitian and (shifted) Hermitian indefinite matrices as important special cases. Results for general normal matrices can be found in previous works . While these can be applied to general analytic functions, as long as f ( A ) is defined, most of the results fail to be insightful in practical situations, as they typically depend on quantities that are very hard or impossible to compute in practice (e.g., because they require knowledge of the complete spectrum of the matrix A ) or are very pessimistic and therefore do not capture the actual quantitative decay behavior well.…”

“…[8][9][10][11][12] Further results on other classes of functions can be found in previous works 9,10,13 ; see also the recent survey by Benzi. 14 There is much interest in finding bounds or estimates for the off-diagonal entries of matrix functions because these allow us to efficiently find sparse approximations of quantities of interest in a variety of areas such as Markov chain queuing models 15,16 and quantum dynamics. 17 Under certain conditions, for example, when the decay behavior is independent of the matrix size n for a family of matrices A n ∈ C n×n , the knowledge of sharp decay bounds even allows the design of optimal, linearly scaling algorithms for matrix function computations.…”

“…Results for general normal matrices can be found in previous works. 11,13,14,19 While these can be applied to general analytic functions, as long as f(A) is defined, most of the results fail to be insightful in practical situations, as they typically depend on quantities that are very hard or impossible to compute in practice (e.g., because they require knowledge of the complete spectrum of the matrix A) or are very pessimistic and therefore do not capture the actual quantitative decay behavior well. Recently, 12 sharp and practicable decay bounds were obtained for functions that are analytic on a connected set containing the field of values of A.…”