  Matrix pencils, or pairs of matrices, may be used in a variety of
applications. In particular, a pair of matrices (E,A) may be interpreted as the
differential equation E x' + A x = 0. Such an equation is invariant by changes
of variables, or linear combination of the equations. This change of variables
or equations is associated to a group action. The invariants corresponding to
this group action are well known, namely the Kronecker indices and divisors.
Similarly, for another group action corresponding to the weak equivalence, a
complete set of invariants is also known, among others the strangeness.
  We show how to define those invariants in a directly invariant fashion, i.e.
without using a basis or an extra Euclidean structure. To this end, we will
define a reduction process which produces a new system out of the original one.
The various invariants may then be defined from operators related to the
repeated application of the reduction process. We then show the relation
between the invariants and the reduced subspace dimensions, and the relation
with the regular pencil condition. This is all done using invariant tools only.
  Making special choices of basis then allows to construct the Kronecker
canonical form. In a related manner, we construct the strangeness canonical
form associated to weak equivalence.
