parameterization of equitable matrices

A $n\times n$ matrix is equitable if and only if it can be expressed in the form

$$m_{ij}=\exp ({\lambda }_i-{\lambda }_j)$$

for real numbers ${\lambda }_1,{\lambda }_2,\mathrm{\dots },{\lambda }_n$ with ${\lambda }_1=0$.

Assume that $m_{ij}$ are the entries of an equitable matrix.

Since all the elements of an equitable matrix are positive by definition, we can write

$$m_{ij}=\exp {\mu }_{ij}$$

with the quantities ${\mu }_{ij}$ being real numbers (which may be positive, negative or zero).

In terms of this representation, the defining identity for an equitable matrix becomes

$${\mu }_{ik}={\mu }_{ij}+{\mu }_{jk}$$

Since this comprises a system of linear equations for the quantities ${\mu }_{ij}$, we could solve it using the usual methods of matrix theory. However, for this particular system of linear equations, there is a much simpler approach.

Consider the special case of the identity when $i=j=k$:

$${\mu }_{ii}={\mu }_{ii}+{\mu }_{ii}.$$

This simplifies to

$${\mu }_{ii}=0.$$

In other words, all the diagonal entries are zero.

Consider the case when $i=k$ (but does not equal $j$).

$${\mu }_{ij}+{\mu }_{ji}={\mu }_{ii}$$

By wat we have just shown, the right hand side of this equation equals zero. Hence, we have

$${\mu }_{ij}=-{\mu }_{ji}.$$

In other words, the matrix of $\mu$’s is antisymmetric.

We may express any entry in terms of the $n$ entries ${\mu }_{i1}$:

$${\mu }_{ij}={\mu }_{i1}+{\mu }_{1j}={\mu }_{i1}-{\mu }_{j1}$$

We will conclude by noting that if, given any $n$ numbers ${\lambda }_i$ with ${\lambda }_1=0$, but the remaining $\lambda$’s arbitrary, we define

$${\mu }_{ij}={\lambda }_i-{\lambda }_j,$$

then

$${\mu }_{ij}+{\mu }_{jk}={\lambda }_i-{\lambda }_j+{\lambda }_j-{\lambda }_k={\lambda }_i-{\lambda }_k={\mu }_{ik}$$

Hence, we obtain a solution of the equations

$${\mu }_{ik}={\mu }_{ij}+{\mu }_{jk}.$$

Moreover, by what we what we have seen, if we set ${\lambda }_i={\mu }_{i1}$, all solutions of these equations can be so described.

Q.E.D.

Title	parameterization of equitable matrices
Canonical name	ParameterizationOfEquitableMatrices
Date of creation	2013-03-22 14:58:36
Last modified on	2013-03-22 14:58:36
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	6
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 15-00