Reduction of endomorphisms

Historically, linear algebra arose from the need to provide a solid foundation for the study of systems of linear equations, but also from the need to grasp what survived from Euclid's geometry, once the effect of translations had been erased and, eventually, the idea of distance forgotten. The reduction of endomorphisms only appeared later, and it was in his examination of differential equations with regular singularities (Fuchs theory) that C. Jordan tackled the reduction that would bear his name.

Linear algebra is gradually developing into a speciality worthy of interest in its own right, and is becoming, in the elementary sense of the term, the "science" that deals with matrices or vector spaces and linear applications between these vector spaces. The basic objectives are reduced, roughly speaking, to the examination of four, or even five, main equivalence relations defined between matrices. These are:

• r-equivalence (A = PBQ);

• of PG-equivalence (A = PB), which is the basis of the first of the historical sources mentioned above (PG as Gaussian pivot);

• similarity (A = PBP ^–1 ), which is the subject of our study;

• congruence (A = PB ^t P).

Finally, another relationship establishes certain links between similarity and congruence; it is given by the orthogonal similarity A = OBO ^–1 = OB ^t O.

From now on, the aim will be to identify the criteria for belonging or not belonging to a given equivalence class, although it will not always be possible to give an explicit description of these classes. The presentation adopted here makes free use of the language of operating groups, each class being an orbit under the action of the group appropriate to the situation.

• Two aspects need to be taken into account when it comes to silitude.

A classical approach consists, once a matrix A of order n with coefficients in the body $\mathbb{K}$ has been chosen, in finding in its similarity class a matrix with a simple form (diagonal, when possible, or, failing that, tridiagonal or triangular, etc.), and we then say reduce it, then find an element of the linear group GL(n, $$