# equitable matrix

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

###### Definition 1.

An $n\times n$ matrix $M=(m_{ij})$ is an equitable matrix if all $m_{ij}$ are positive, and $m_{ij}=m_{ik}m_{kj}$ for all $i,j,k=1,\ldots,n$.

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

## Examples

1. 1.

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

 $\begin{pmatrix}1&\cdots&1\\ \vdots&\ddots&\vdots\\ 1&\cdots&1\end{pmatrix}.$

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

2. 2.

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

 $\begin{pmatrix}1&a\\ 1/a&1\end{pmatrix},$

and

 $\begin{pmatrix}1&a&ab\\ 1/a&1&b\\ 1/ab&1/b&1\end{pmatrix},$

where $a,b,c>0$.

## Properties

1. 1.

A $n\times n$ matrix $M=(m_{ij})$ 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 $\lambda_{1}=0$. (proof. (http://planetmath.org/ParameterizationOfEquitableMatrices))

2. 2.

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

 $m_{ij}=\frac{m_{1j}}{m_{1i}}.$
3. 3.

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

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

where $\operatorname{exp}$ is the matrix exponential.

4. 4.

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

5. 5.

If $M$ is an $n\times n$ equitable matrix and $s\colon\{1,\ldots,r\}\to\{1,\ldots,n\}$ is a mapping, then

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

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

See [1] for further properties and references.

## References

Title equitable matrix EquitableMatrix 2013-03-22 14:58:28 2013-03-22 14:58:28 matte (1858) matte (1858) 11 matte (1858) Definition msc 15-00 HadamardProduct