  In this paper many classes of sets of matrices with entries in F (F=R, F=C,
F=H) are introduced. Each class with the corresponding topology determines a
real analytical, complex or symplectic manifold for F=R, F=C or F=H
respectively. Any such family is called to be a set of canonical forms of
matrices. The construction of such canonical forms of matrices is introduced
inductively. First basic canonical forms are introduced, and then two
operations for obtaining new canonical forms by using the old canonical forms
are introduced. All such manifolds have the property that each of them can be
decomposed into cells of type $F^i$.
