Eigenvalues. Characteristic polynomial. Minimal polynomial

The existence of a finite family of scalars (in this case, eigenvalues) associated with a matrix, or more precisely its similarity class, is a remarkable phenomenon. We'll approach it mathematically from several points of view. This phenomenon can also be seen in certain physical situations. Trubowitz springs provide an easy example: the study of the motion of n identical masses, moving on a circular rail and linked by equally identical springs. The existence and determination of the so-called pure states of the system (cases where the masses all oscillate with the same frequency, and which are finite in number), are in fact intimately linked to the existence and study of the eigenvalues of a matrix naturally associated with the system (cf. §