@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.}
}