Last week we had a fascinating seminar by Tom Leinster, from Glasgow, with the title “Entropy, diversity and magnitude”. The abstract is here, and certainly suggested something interesting in prospect.
I’ll introduce the topic with a modification of one of Tom’s examples. Suppose that you are an ecologist studying a wood which contains oak trees and two very closely related species of pine trees. How many different kinds of tree are there? You may well feel that the number is smaller than if the trees were oak, pine, and hornbeam, say: slightly more than two, but certainly less than three.
We were shown a way to measure this, as well as the size of a finite category (its Euler characteristic), the size of a metric space, and several other examples. The technique depends on the following result about real square matrices:
Let A be a real square matrix. Suppose there exist a row vector v and a column vector w such that vA and Aw are each equal to the appropriate all-1 vector. Then the sum of the entries of v is equal to the sum of the entries of w (and so is independent of the choice of v and w).
The proof of this is an easy exercise.
The number defined in this result will be called the size of the matrix A, written |A| (not to be confused with the determinant or the modulus). The conditions of the theorem are satisfied by almost all matrices, and in particular by all invertible matrices; if A is invertible then its size is the sum of the entries in its inverse. But the size can exist for singular matrices too: for example, the size of the all-1 2×2 matrix is 1.
Note that the size may be zero or negative.
Thus, if we choose to represent n completely separate objects by a n×n identity matrix (each diagonal entry 1, each off-diagonal entry zero), the size of the matrix is n, the cardinality of the set. But if the objects are related in some way, indicated by positive off-diagonal entries, we expect the size to be smaller; for the all-1 matrix, the size is 1, since the objects are effectively the same.
One amusing specialisation is to the case of a finite category, where the (i,j) entry of the matrix is the number of homomorphisms from the i-th object to the j-th. The size of the matrix is the Euler characteristic of the category, a standard measure of “size” in this context.
A very interesting case involves metric spaces, where we take the (i,j) entry to be a negative exponential of the distance between the i-th and j-th points. This example can be used to encourage the ecological example with which we began.
A degree of technical material was hidden away. For example, using a limiting process, we get a well-defined notion of size for a compact metric space. Rather than a single number, we can look at the function of t giving the size of the space when the metric is scaled by a factor of t. For a compact convex set in the plane, it is conjectured that the resulting function is quadratic, with the coefficients of the linear and quadratic terms being (up to scaling) the perimeter and area of the set.
The final section of the talk was on related measures of entropy and spread, potentially of interest to the biologists.
Surely there is much more fun to be had here …