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. $\Vert Y_n\Vert _1=o(n)$.

Under these assumptions, the functional $\phi :C_c(\mathbb C)\rightarrow \mathbb C$ identifying the spectral distribution in the sense of Definition 3.1 is the same for $\{X_n\}_n$ and $\{X_n+Y_n\}_n$. This result should be regarded as a useful addendum to the theory of GLT sequences. In practice, it turns out to be an effective tool for computing the spectral distribution of GLT sequences formed by perturbed Hermitianmatrices. In this respect, we point out that the matrix-sequences coming from a DE discretization are often GLT sequences formed by symmetric or nearly symmetric matrices, at least in the case where the higher-order differential operator of the considered DE is symmetric (self-adjoint).