The companion matrix C of a (monic) polynomial p is uniquely
associated with the representation of p in terms of monomials,
and p is the characteristic polynomial of C.
An equivalence transformation of C corresponds to a different
representation of p, for example in a (generalized) Newton form.
In the literature, this basic fact has been used for various
algorithmic purposes, e.g. for the refinement of given approximations
of the zeros of p using tools from Numerical Linear Algebra.
In this talk we take a different point of view on companion matrices.
In Numerical Analysis, for instance, discretization methods are
often characterized by certain polynomials resp. their companion
matrices C. A typical example are linear multistep methods for ODEs.
In this way, theoretical questions like stability issues can be formulated
in terms of properties of C. In this talk we demonstrate
that the successs of this approach crucially depends on the choice of
basis resp. an appropriate canonical form for C. Important special
cases are the Bidiagonal and the Bidiagonal-Frobenius form which
are associated with generalized Newton representations for p.
In contrast to the Jordan form, these canonical forms are
typically continuous w.r.t. certain parameters of the problem or
the method considered, which make them a very useful and flexible
tool for theoretical purposes, e.g. in stability theory.
Some simple examples for this type of analysis will be given.