  During the whole of 1874, Camille Jordan and Leopold Kronecker quar- relled
vigorously over the organisation of the theory of bilinear forms. That theory
promised a "general" and "homogeneous" treatment of numerous questions arising
in various 19th-century theoretical contexts, and it hinged on two theorems,
stated independently by Jordan and Weierstrass, that would today be considered
equivalent. It was, however, the perceived difference between those two
theorems that sparked the 1874 controversy. Focusing on this quarrel allows us
to explore the algebraic identity of the polynomial practices of the
manipulations of forms in use before the advent of structural approaches to
linear algebra. The latter approaches identified these practices with methods
for the classification of similar matrices. We show that the prac- tices --
Jordan's canonical reduction and Kronecker's invariant computation -- reflect
identities inseparable from the social context of the time. Moreover, these
practices reveal not only tacit knowledge, local ways of thinking, but also --
in light of a long history tracing back to the work of Lagrange, Laplace, Cau-
chy, and Hermite -- two internal philosophies regarding the significance of
generality which are inseparable from two disciplinary ideals opposing algebra
and arithmetic. By interrogating the cultural identities of such practices,
this study aims at a deeper understanding of the history of linear algebra
without focusing on issues related to the origins of theories or structures.
