equitable matrix

Equitable matrices have been used in economics and group theory[1].

Definition 1.

An $n\mathrm{\times }n$ matrix $M\mathrm{=}\mathrm{(}{m}_{i\mathit{}j}\mathrm{)}$ is anequitable matrix if all $m_{i\mathit{}j}$ are positive, and$m_{i\mathit{}j}\mathrm{=}{m}_{i\mathit{}k}\mathit{}{m}_{k\mathit{}j}$ for all $i\mathrm{,}j\mathrm{,}k\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}n$.

Setting $i=j=k$ yields $m_{ii}=m_{ii}{m}_{ii}$ so diagonal elements ofequitable matrices equal $1$. Next, setting $i=j$ yields$m_{ii}=m_{ik}{m}_{ki}$, so $m_{ik}=1/m_{ki}$.

Examples

1. An example of an equitable matrix of order $n$ is

   This example shows that equitable matrices exist for all $n$.

2. The most general equitable matrix of orders $2$ and $3$ are

   and

   where $a,b,c>0$.

Properties

1. A $n\times n$ matrix $M=(m_{ij})$ is equitable if andonly 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 ${\lambda }_1=0$. (proof. (http://planetmath.org/ParameterizationOfEquitableMatrices))

2. An equitable matrix is completely determined by its first row.If $m_{1i}$, $i=1,\mathrm{\dots },n$ are known, then

   $$m_{ij}=\frac{m_{1j}}{m_{1i}}.$$

3. If $M$ is an $n\times n$ equitable matrix, then

   $$\exp (M)=I+\frac{e^n-1}{n}M,$$

   where $\exp$ is the matrix exponential.

4. Equitable matrices form a group under the Hadamard product [1].

5. If $M$ is an $n\times n$ equitable matrix and $s:\{1,\mathrm{\dots },r\}\to \{1,\mathrm{\dots },n\}$is a mapping, then

   $$K_{ab}=M_{s(a)s(b)},a,b=1,\mathrm{\dots },r$$

   is an equitable $r\times r$ matrix. In particular, striking the $l$:th row and column in anequitable matrix yields a new equitable matrix.

See [1] for further properties and references.