Tools for Determining the Asymptotic Spectral Distribution of non-Hermitian Perturbations of Hermitian Matrix-Sequences and Applications

Abstract

Given a sequence of matrices (matrix-sequence) {Xn}, with Xn Hermitian of size dn tending to infinity, we consider the sequence {Xn + Yn}, where {Yn} is an arbitrary (non-Hermitian) perturbation of {Xn}. We prove that {Xn + Yn} has an asymptotic spectral distribution if: {Xn} has an asymptotic spectral distribution, the spectral norms \({\|X_n\|,\|Y_n\|}\) are uniformly bounded with respect to n, and {Yn} satisfies a trace-norm assumption. Furthermore, under the above assumptions, the functional ϕ identifying the asymptotic spectral distribution is the same for {Xn + Yn} and {Xn}. We mention some examples of applications, including the case of matrix-sequences with asymptotic spectral distributions described by matrix-valued functions and the approximation by \({\mathbb{Q}_k}\) Finite Element methods of convection-diffusion equations.