Localization of Matrix Factorizations

Abstract

Matrices with off-diagonal decay appear in a variety of fields in mathematics and in numerous applications, such as signal processing, statistics, communications engineering, condensed matter physics, and quantum chemistry. Numerical algorithms dealing with such matrices often take advantage (implicitly or explicitly) of the empirical observation that this off-diagonal decay property seems to be preserved when computing various useful matrix factorizations, such as the Cholesky factorization or the QRfactorization. There is a fairly extensive theory describing when the inverse of a matrix inherits the localization properties of the original matrix. Yet, except for the special case of band matrices, surprisingly very little theory exists that would establish similar results for matrix factorizations. We will derive a comprehensive framework to rigorously answer the question when and under which conditions the matrix factors inherit the localization of the original matrix for such fundamental matrix factorizations as the LU-, QR-, Cholesky, and Polar factorization.

Decay Bounds for Functions of Hermitian Matrices with Banded or Kronecker Structure

“…A related but quite distinct line of research concerned the study of inverse-closed matrix algebras, where the decay behavior in the entries of a (usually infinite) matrix A is "inherited" by the entries of A −1 . Here we mention [33], where it was observed that a similar decay behavior occurs for the entries of f (A) = A −1/2 , as well as [2,3,26,27,35], among others.The study of the decay behavior for general analytic functions of banded matrices, including the important case of the matrix exponential, was initiated in [6,32] and continued for possibly nonnormal matrices and general sparsity patterns in [7]; further contributions in these directions include [4,16,38,42]. Collectively, these papers have largely elucidated the question of when one can expect exponential decay in the entries…”

“…The interest for the decay behavior of matrix functions stems largely from its importance for a number of applications, including numerical analysis [6,13,16,17,22,40,46], harmonic analysis [2,26,33], quantum chemistry [5,11,37,42], signal processing [35,43], quantum information theory [14,15,23], multivariate statistics [1], queuing models [9,10], control of large-scale dynamical systems [29], quantum dynamics [25], random matrix theory [41], and others. The first case to be analyzed in detail was that of f (A) = A −1 ; see [17,18,22,34].…”

Contraction and Optimality Properties of an Adaptive Legendre–Galerkin Method: The Multi-Dimensional Case

“…Proof. We proceed extending the idea of [3,Theorem 4.1] to the present infinite-dimensional setting (see also [21,23] for similar results). Recall that S η has been normalized so that (S η ) i,i, = 1 for all i ∈ K. Here S η = LL T is the Cholesky factorization of S η (with L lower triangular infinite dimensional matrix).…”

“…The authors would like to thank Michele Benzi for helpful discussions and for pointing to [23]. The first and third author have been partially supported by the Italian research grant Prin 2012 2012HBLYE4 004 "Metodologie innovative nella modellistica differenziale numerica".…”

Block diagonal dominance of matrices revisited: Bounds for the norms of inverses and eigenvalue inclusion sets

“…In this context it is then of great practical interest to know a priori how many and which of these entries can be discarded as insignificant. Many authors have therefore studied decay rates for different matrix classes and functions of matrices; see, e.g., [2,4,5,6,7,10,14,18]. For an excellent survey of the current state-of-the-art we refer to [1].An important example in this context is given by the (nonsymmetric) diagonally dominant matrices, and in particular the diagonally dominant tridiagonal matrices, which were studied, e.g., in [15,16].…”