Finite determinacy of matrices
Let f be a power series (in several variables) or a C^\infty-smooth function. In many cases just a finite part of Taylor expansion is enough to determine f up to the change of coordinates. Alternatively, the deformations of f by terms of high enough orders are trivial. This phenomenon is called the finite determinacy.
An immediate application is the algebraization: f has a polynomial representative.
More generally, for maps of smooth spaces the finite determinacy (under various group-actions) has been intensively studied for about 50 years (by Mather, Tougeron, Arnol'd, Wall and many others).