Spectral Distribution of Sequences of Perturbed Hermitian Matrices

Abstract

In this chapter, we present a general result from [65, 68], which is useful for computing the spectral distribution of matrix-sequences of the form \(\{X_n+Y_n\}_n\), where \(X_n\) is Hermitian and \(Y_n\) is a ‘small’ perturbation of \(X_n\). More precisely, we prove that \(\{X_n+Y_n\}_n\) has an asymptotic spectral distribution if: 1. \(\{X_n\}_n\) is a sequence of Hermitian matrices possessing an asymptotic spectral distribution; 2. the spectral norms \(\Vert X_n\Vert ,\Vert Y_n\Vert \) are uniformly bounded with respect to n; 3. \(\Vert Y_n\Vert _1=o(n)\).