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Metonymy and Metaphor in Mathematics

5. Metaphors from modern mathematics:

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

Two $n\times n$ matrices A and B are similar if there is an invertible $n\times n$ 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 $n\times n$ matrix A with complex entries is similar to a matrix in Jordan normal form. This is a matrix of the form

\begin{displaymath}\left ( \begin{array}{cccccc} L_1 &0 &0 &... &0 &0\\ 0 &L_2 ... ...&... &0 &L_{p-1} &0\\ 0 &0 &... &0 &0 &L_p \end{array}\right )\end{displaymath}

where $L_1,\dots, L_p$ 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

\begin{displaymath}\left ( \begin{array}{ccccccc} \lambda &0 &... &...&... &... ... ...&\lambda &0\\ 0 &0 &0 &... &0 &1 &\lambda \end{array}\right )\end{displaymath}

for some complex number $\lambda$.

The metaphier. The structure theorem for finite abelian groups says that a finite abelian G is isomorphic to a direct sum of cyclic groups ${\bf Z}_{\textstyle r_i}$, 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 ${\bf Z}_2\oplus{\bf Z}_2\oplus{\bf Z}_3$ or ${\bf Z}_4\oplus{\bf Z}_3$. There are no other possibilities. This generalizes to a structure theorem for modules over a principal ideal domain. ${\bf Z}$ and the ring ${\bf C}[X]$ of complex-valued polynomials are both p.i.d.s; an abelian group is naturally a ${\bf Z}$-module, with $n\cdot g = g + \cdots +g$ (n times).

The metaphor. Use A to make ${\bf C}^n$ into a ${\bf C}[X]$-module by letting $p(X)\cdot {\bf v} = p(A){\bf v}$ so $(X^2 - 3iX +2)\cdot {\bf v} = AA{\bf v} -3iA{\bf v}+2{\bf v}$, etc. The analogue of ${\bf Z}$ mod the multiples of ri (a prime power) is ${\bf C}[X]$ mod the multiples of $(X-\lambda)^d$, since in ${\bf C}[X]$ the irreducible polynomials are linear.

By the structure theorem, ${\bf C}^n$ splits up into subspaces

\begin{displaymath}{\bf C}^n = E_1\oplus\cdots \oplus E_p,\end{displaymath}

and restricted to Ei the module structure is the same as the action of ${\bf C}[X]$ on ${\bf C}[X]/((X-\lambda)^d)$, for some complex $\lambda$. A basis for the quotient is

\begin{displaymath}1,X-\lambda, (X-\lambda)^2,\dots,(X-\lambda)^{d-1}.\end{displaymath}

With respect to this basis the matrix of A, which corresponds to multiplication by X, comes out as follows:

\begin{displaymath}\begin{array}{clll} 1 &\rightarrow& X& = \lambda\cdot 1 +1\cd... ...& = \lambda(X-\lambda)^{d-1} ~~ mod~~ (X-\lambda)^d \end{array}\end{displaymath}

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.)

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