II. The Jordan normal form and the structure of abelian groups
Two
matrices A and B are similar if there is an
invertible
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.
Any
matrix A with complex entries is similar to
a matrix in Jordan normal form. This is a matrix of the
form
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
cyclic groups
,
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
or
.
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,
with
(n times).
The metaphor. Use A to make
into a
-module
by letting
so
,
etc. The analogue of
mod
the multiples of ri (a prime power) is
mod
the multiples of
,
since in
the irreducible
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.)
5. Metaphors from modern mathematics:
II. The Jordan normal form and the structure of abelian groups