Frédéric Brechenmacher - Centre Alexandre Koyré
An introduction to the theory of canonical matrices. Turnbul et Aitken , 1932.
We have preferred to follow the lead of Cullis, who develops the theory in terms of the structure and properties of matrices – in matrix idiom, as it were, rather than in terms of bilinear and quadratic forms, or of linear substitutions. [...], we have, at the same time, aimed at a certain compactness in the formulae and demonstrations. This has been achieved in the first place by a systematic use of the matrix notation […]. The theory of canonical matrices is concerned with the systematic investigation of types of transformation which reduce matrices to the simplest and most convenient shape. […] in the second place, by confining the contents of each chapter almost entirely to general theorems, and by relegating corollaries and applications to the interspersed sets of examples. These examples are intended to serve not so much as exercises many being quite easy, but rather as points of relaxation, and running commentary ; they will, however, be found to contain many well-known and important theorems, which the notation establishes in the minimum of space. […] The reader already familiar with the theory will also observe that certain established methods of dealing with the subject have hardly been touched upon, notably the methods of Weierstrass and Darboux, the theory of regular minors of the determinants and the treatment of quadratic forms by the method of Kronecker. We have, in fact, allowed ourselves a free hand in dealing with the results of earlier writers, in the belief that the outcome would prove to be an easier approach to a subject that has often failed to win affection; and the methods of H.J.S. Smith, Sylvester, Frobenius and Dickson proved in themselves quite adequate without the inclusion of other parallel theories. [Turnbul et Aitken, 1932, 1].
Plan du traité :
Definitions and fundamental properties of matrices.
Elementary Transformations. Bilinear and Quadratic Forms.
The canonical reduction of Equivalent matrices
Subgroups of the Group of equivalent transformations.
A Rational Canonical Form for the Collineatory group.
The Classical Canonical Form for the Collineatory Group.