  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.
