metonymy5 **Metonymy and Metaphor in Mathematics**

## 5. Metaphors from modern mathematics:

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 *n*, arranged along the diagonal, and the 0s represent appropriate blocks of zeroes. Furthermore each *L*_{i} 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 *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 *E*_{i} 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 *L*_{i} 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.)

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