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*^{-1}*AS*. 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

where are square blocks of various sizes adding up to

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 *r*_{i} 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 *r*_{i} (a prime power) is
*mod*
the multiples of
,
since in
the irreducible
polynomials are linear.

By the structure theorem,
splits up into subspaces

and restricted to

With respect to this basis the matrix of

giving exactly the block

(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.)

- 1. Metonymy and metaphor
- 2. How to recognize
mathematical metonymy
- 3. How to recognize
a mathematical metaphor
- 4. Metaphors from modern mathematics:

I. Fourier analysis and the dot-product - 5. Metaphors from modern mathematics:

II. The Jordan normal form and the structure of abelian groups

© copyright 1999, American Mathematical Society.