Block-diagonalisation of matrices and operators

TL;DR

This paper presents a constructive method for asymptotically block-diagonalizing families of matrices and operators, facilitating the analysis of their spectral properties as parameters tend to zero.

Contribution

It introduces a new scheme for decomposing matrix families into blocks based on eigenvalues, extending to parameter-dependent matrices and operators on Hilbert spaces.

Findings

Effective block-diagonalization scheme for matrix families

Extension to parameter-dependent matrices and operators

Applicable to spectral analysis in various contexts

Abstract

In this short note we deal with a constructive scheme to decompose a continuous family of matrices $A(\rho)$ asymptotically as $\rho\to0$ into blocks corresponding to groups of eigenvalues of the limit matrix A(0). We also discuss the extension of the scheme to matrix families depending upon additional parameters and operators on Hilbert spaces.