Uppsala University
Department of Information Technology

### Technical Report 2018-004

# Non-Hermitian Perturbations of Hermitian Matrix-Sequences and Applications to the Spectral Analysis of Approximated PDEs

### Giovanni Barbarino and Stefano Serra-Capizzano

February 2018

**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_{λ} 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 *\|·\|*_{2} being the Schatten 2 norm, then *{X*_{n}+Y_{n}}_{n}sim_{λ} 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.

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