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
where are square blocks of various sizes adding up to n, arranged along the diagonal, and the 0s represent appropriate blocks of zeroes. Furthermore each Li has 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
and restricted to Ei the module structure is the same as the action of on , for some complex . A basis for the quotient is
With respect to this basis the matrix of A, which corresponds to multiplication by X, comes out as follows:
giving exactly the block Li described above.
(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.)