II. The Jordan normal form and the structure of abelian groups
matrices A and B are similar if there is an
matrix S such that
B=S-1AS. This means
that A and B are ``doing the same thing'' but seen with respect
to two different sets of basis vectors.
matrix A with complex entries is similar to
a matrix in Jordan normal form. This is a matrix of the
for some complex number .
The metaphier. The structure theorem for finite abelian groups
says that a finite abelian G is isomorphic to a direct sum of
where ri is a
power of a prime, and that this decomposition is unique up to
ordering. For example, an abelian group of order 12 must be
isomorphic to either
There are no other
possibilities. This generalizes to
a structure theorem for modules over a principal ideal domain.
and the ring
of complex-valued polynomials
are both p.i.d.s; an abelian group is naturally a -module,
The metaphor. Use A to make
etc. The analogue of
the multiples of ri (a prime power) is
the multiples of
polynomials are linear.
By the structure theorem,
splits up into subspaces
(For more details see a text like Hartley and Hawkes, Rings, Modules and Linear Algebra,Chapman & Hall, London, New York, 1970 or Serge Lang, Algebra, Addison-Wesley 1971.)