Diagonalisation schemes and applications

TL;DR

This paper develops perturbation theory techniques for matrices, focusing on diagonalisation schemes to derive asymptotic eigenvalue expansions, with applications across differential equations, hyperbolic polynomials, and symbolic hierarchies.

Contribution

It introduces constructive diagonalisation schemes for asymptotic eigenvalue analysis and demonstrates their applications in various mathematical frameworks.

Findings

Asymptotic expansions for eigenvalues and eigenprojections derived

Applications to differential equations and hyperbolic systems demonstrated

Frameworks for symbolic and pseudo-differential analysis developed

Abstract

These notes develop aspects of perturbation theory of matrices related to so-called diagonalisation schemes. Primary focus is on constructive tools to derive asymptotic expansions for small/large parameters of eigenvalues and eigenprojections of families of matrices depending upon real/complex parameters. Applications of the schemes in different frameworks --including hyperbolic polynomials, asymptotic integration of ordinary differential equations, diagonalisation within symbolic hierarchies and pseudo-differential decoupling of hyperbolic-parabolic coupled systems-- are also discussed and references to further applications given.