Autour de la d\'ecomposition de Dunford r\'eelle ou complexe. Th\'eorie spectrale et m\'ethodes effectives

TL;DR

This paper reviews methods for Dunford's spectral decomposition over real or complex fields, focusing on effective algorithms for cases where eigenvalues are unknown, including Newton-Raphson and Chinese Remainder techniques.

Contribution

It analyzes and compares effective methods for Dunford's decomposition, especially when eigenvalues are not directly accessible, emphasizing Chevalley's approach and diagonalizability testing.

Findings

Newton-Raphson method converges quadratically for diagonalizable components.

Chinese Remainder-based algorithms provide effective spectral projectors.

A Sturm-based test for diagonalizability in real matrices is proposed.

Abstract

These notes are not intended to substitute for a course in linear algebra on reduction of endomorphisms nor an exhaustive presentation of the Dunford's decomposition. We will limit ourselves to the case where the base is R or C, and the purpose of this presentation is to make an inventory of the various Dunford's decomposition methods. When the eigenvalues are known with their exact values, decomposition into simple elements of the inverse of a polynomial annihilator provides us the spectral projectors and a fortiori the expected decomposition. The most difficult case occurs when the spectrum of the endomorphism is not at our disposal, which is a common situation when the dimension of the vector space is greater than 4. The Newton-Raphson method then comes to the rescue to provide a sequence which converges quadratically to diagonalizable component. While this method is very popular quite effective regardless of the size matrix studied, but it leaves us hungry. Indeed, we know that Dunford components are polynomials in the matrix and would know these generator polynomials. The good news is that effective method using the Chinese lemma there and it was introduced by Chevalley in the fifty years of the century last. I will focus on this method which was mentioned in an article of Danielle Couty, Jean Esterle and Rachid Zarouf, detailing evidence of the algorithm where the characteristic polynomial is divided on the body base, then I will detail the actual case is a more subtle situation requiring further study. A reminder of the semi-simple endomorphisms was introduced to justify the importance of finding an effective method for testing diagonalisability in Mn (R) when no eigenvalues of the endomorphism studied. To achieve this I have proposed as the Sturm verification tool diagonalisabilit\'e in R.