Analytic, Differentiable and Measurable Diagonalizations in Symmetric Lie Algebras

TL;DR

This paper extends perturbation theory to symmetric Lie algebras, enabling analytic and differentiable diagonalizations for structured matrices, unifying various matrix decompositions within a Lie algebra framework.

Contribution

It generalizes classical matrix diagonalization results to semisimple orthogonal symmetric Lie algebras, providing conditions for smooth diagonalizations of structured matrix paths.

Findings

Established conditions for smooth diagonalizations in symmetric Lie algebras

Unified various matrix decompositions under a Lie algebra framework

Extended perturbation theory to structured matrix families

Abstract

We generalize several important results from the perturbation theory of linear operators to the setting of semisimple orthogonal symmetric Lie algebras. These Lie algebras provide a unifying framework for various notions of matrix diagonalization, such as the eigenvalue decomposition of real symmetric or complex Hermitian matrices, and the real or complex singular value decomposition. Concretely, given a path of structured matrices with a certain smoothness, we study what kind of smoothness one can obtain for the corresponding diagonalization of the matrices.