  The legacy of Jordan's canonical form on Poincar\'e's algebraic practices.
This paper proposes a transversal overview on Henri Poincar\'e's early works
(1878-1885). Our investigations start with a case study of a short note
published by Poincar\'e on 1884 : "Sur les nombres complexes". In the
perspective of today's mathematical disciplines - especially linear algebra -,
this note seems completely isolated in Poincar\'e's works. This short paper
actually exemplifies that the categories used today for describing some
collective organizations of knowledge fail to grasp both the collective
dimensions and individual specificity of Poincar\'es work. It also highlights
the crucial and transversal role played in Poincar\'e's works by a specific
algebraic practice of classification of linear groups by reducing the
analytical representation of linear substitution to their Jordan's canonical
forms. We then analyze in detail this algebraic practice as well as the roles
it plays in Poincar\'e's works. We first provide a micro-historical analysis of
Poincar\'e's appropriation of Jordan's approach to linear groups through the
prism of the legacy of Hermite's works on algebraic forms between 1879 and
1881. This mixed legacy illuminates the interrelations between all the papers
published by Poincar\'e between 1878 and 1885 ; especially between some
researches on algebraic forms and the development of the theory of Fuchsian
functions. Moreover, our investigation sheds new light on how the notion of
group came to play a key role in Poincar\'e's approach. The present paper also
offers a historical account of the statement by Jordan of his canonical form
theorem. Further, we analyze how Poincar\'e transformed this theorem by
appealing to Hermite's
