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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 notepublished by Poincar\'e on 1884 : "Sur les nombres complexes". In theperspective of today's mathematical disciplines - especially linear algebra -,this note seems completely isolated in Poincar\'e's works. This short paperactually exemplifies that the categories used today for describing somecollective organizations of knowledge fail to grasp both the collectivedimensions and individual specificity of Poincar\'es work. It also highlightsthe crucial and transversal role played in Poincar\'e's works by a specificalgebraic practice of classification of linear groups by reducing theanalytical representation of linear substitution to their Jordan's canonicalforms. We then analyze in detail this algebraic practice as well as the rolesit plays in Poincar\'e's works. We first provide a micro-historical analysis ofPoincar\'e's appropriation of Jordan's approach to linear groups through theprism of the legacy of Hermite's works on algebraic forms between 1879 and1881. This mixed legacy illuminates the interrelations between all the paperspublished by Poincar\'e between 1878 and 1885 ; especially between someresearches on algebraic forms and the development of the theory of Fuchsianfunctions. Moreover, our investigation sheds new light on how the notion ofgroup came to play a key role in Poincar\'e's approach. The present paper alsooffers a historical account of the statement by Jordan of his canonical formtheorem. Further, we analyze how Poincar\'e transformed this theorem byappealing to Hermite's

Autour de pratiques alg\'ebriques de Poincar\'e : h\'eritages de la r\'eduction de Jordan