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

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

Bibliography

1

H. Eves, Elementary Matrix Theory, Dover publications, 1980.