Weierstrass and the theory of matrices

The following essay is the third in a series devoted to the history of the theory of matrices.1 In [1975 a] I related Cauchy's important memoir on the characteristic roots of a real quadratic form to the mechanical investigations of Lagrange and Laplace. The origins and development of t e symbolical algebra matrices are traced in [1977]. Now I present the keystone of the history: the theory of canonical matrix forms and, in particular, the work of Karl Weierstrass. The emphasis upon Weie strass reflects my conclusion that through his theory of ele entary divisors [1868] and its influence upon other mathematicians, he, more than anyone, was respo sible for the emerge ce of the theory of matrices as a cohere t, substantial b anch of twentieth-century mathematics. In view of the importa ce of Weierstrass' memoir on elementary divisors, special attention has been given to the background against which it was created and to the mathematicians who influenced hi (Sectio s 2-4). In [1868] and a preliminary memoir [1858] Weierstrass demonstrated more than theorems. He also demonstrated the possibility and desirability of a ore rigorous approach to algebraic analysis that did not rest content with the prevailing tendency to reason vaguely in terms of the "general" case. Weierstrass' memoir thus served as a paradigm for further research by his colleagues and students such as Kronecker and Frobeni s. It was by virtue of the influence of his work on others that Weierstrass played such a key role in the evolution of the theory of matrices. Sec ion 6 is entirely devoted to the fundamental contributions of Frobenius while in Section 7 I establish the domi ant role played y the Berlin school (Wei rstrass, Kronecker, Frobenius) in the dev lopm nt, application, syste-

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