Autour de la décomposition de Dunford réelle ou complexe. Théorie spectrale et méthodes effectives
Résumé
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 field is R or C, and the purpose of the presentation is to make an inventory of various methods of Dunford decomposition . 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's method then comes to the rescue to provide a sequence which converges quadratically to diagonalizable component. While this method very popular is quite effective for all size of the matrix studied, but it leaves us hungry. In effect, we know that Dunford components are polynomials in the matrix and would like know these generator polynomials. The good news is that effective method using the Chinese lemma exists and it was introduced by Chevalley in the fifties of the last century. I will focus on this method which was mentioned in an article of Danielle Couty, Jean Esterle and Rachid Zarouf, by detailing evidence of the algorithm in the case where the characteristic polynomial is divided on the field base, then I will detail the real case which 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 M_n (R) when no eigenvalues of the endomorphism studied. To achieve this I have proposed the Sturm sequence as a verification tool of diagonalisability in R.
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