equitable matrix

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

Definition 1.

An n×n matrix M=(mij) is an equitable matrix if all mij are positive, and mij=mikmkj for all i,j,k=1,,n.

Setting i=j=k yields mii=miimii so diagonal elements of equitable matrices equal 1. Next, setting i=j yields mii=mikmki, so mik=1/mki.


  1. 1.

    An example of an equitable matrix of order n is


    This example shows that equitable matrices exist for all n.

  2. 2.

    The most general equitable matrix of orders 2 and 3 are




    where a,b,c>0.


  1. 1.

    A n×n matrix M=(mij) is equitable if and only if it can be expressed in the form


    for real numbers λ1,λ2,,λn with λ1=0. (proof. (http://planetmath.org/ParameterizationOfEquitableMatrices))

  2. 2.

    An equitable matrix is completely determined by its first row. If m1i, i=1,,n are known, then

  3. 3.

    If M is an n×n equitable matrix, then


    where exp is the matrix exponentialMathworldPlanetmath.

  4. 4.

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

  5. 5.

    If M is an n×n equitable matrix and s:{1,,r}{1,,n} is a mapping, then


    is an equitable r×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.


Title equitable matrix
Canonical name EquitableMatrix
Date of creation 2013-03-22 14:58:28
Last modified on 2013-03-22 14:58:28
Owner matte (1858)
Last modified by matte (1858)
Numerical id 11
Author matte (1858)
Entry type Definition
Classification msc 15-00
Related topic HadamardProduct