@TechReport{ it:2018-004,
author = {Giovanni Barbarino and Stefano Serra-Capizzano},
title = {Non-{H}ermitian Perturbations of {H}ermitian
Matrix-Sequences and Applications to the Spectral Analysis
of Approximated {PDE}s},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2018},
number = {2018-004},
month = feb,
abstract = {This paper concerns the spectral analysis of
matrix-sequences which can be written as a non-Hermitian
perturbation of a given Hermitian matrix-sequence. The main
result reads as follows. Suppose that $X_n$ is a Hermitian
matrix of size $n$ and that $\{X_n\}_n\sim_{\lambda} f$,
i.e., the matrix-sequence $\{X_n\}_n$ enjoys an asymptotic
spectral distribution, in the Weyl sense, described by a
Lebesgue measurable function $f$; if $\|Y_n\|_2 = o(\sqrt
n)$ with $\|\cdot\|_2$ being the Schatten 2 norm, then
$\{X_n+Y_n\}_n\sim_{\lambda} f$.
In a previous paper by Leonid Golinskii and the second
author a similar result was proved, but under the technical
restrictive assumption that the involved matrix-sequences
$\{ X_n\}_n$ and $\{ Y_n\}_n$ are uniformly bounded in
spectral norm. Nevertheless, the result had a remarkable
impact in the analysis of both spectral distribution and
clustering of matrix-sequences arising from various
applications, including the numerical approximation of
partial differential equations (PDEs) and the
preconditioning of PDE discretization matrices. The new
result considerably extends the spectral analysis tools
provided by the former one, and in fact we are now allowed
to analyse variable-coefficient PDEs with unbounded
coefficients, preconditioned matrix-sequences, etc.
A few selected applications are considered, extensive
numerical experiments are discussed, and a further
conjecture is illustrated at the end of the paper.}
}