# 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 $\lambda_{1},\lambda_{2},\ldots,\lambda_{n}$ 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 ParameterizationOfEquitableMatrices 2013-03-22 14:58:36 2013-03-22 14:58:36 rspuzio (6075) rspuzio (6075) 6 rspuzio (6075) Theorem msc 15-00