parameterization of equitable matrices

A n×n matrix is equitable if and only if it can be expressed in the form


for real numbers λ1,λ2,,λn with λ1=0.

Assume that mij are the entries of an equitable matrix.

Since all the elements of an equitable matrix are positive by definition, we can write


with the quantities μij being real numbers (which may be positive, negative or zero).

In terms of this representation, the defining identityPlanetmathPlanetmath for an equitable matrix becomes


Since this comprises a system of linear equations for the quantities μij, 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:


This simplifies to


In other words, all the diagonal entries are zero.

Consider the case when i=k (but does not equal j).


By wat we have just shown, the right hand side of this equation equals zero. Hence, we have


In other words, the matrix of μ’s is antisymmetricMathworldPlanetmath.

We may express any entry in terms of the n entries μi1:


We will conclude by noting that if, given any n numbers λi with λ1=0, but the remaining λ’s arbitrary, we define




Hence, we obtain a solution of the equations


Moreover, by what we what we have seen, if we set λi=μi1, all solutions of these equations can be so described.


Title parameterization of equitable matrices
Canonical name ParameterizationOfEquitableMatrices
Date of creation 2013-03-22 14:58:36
Last modified on 2013-03-22 14:58:36
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Theorem
Classification msc 15-00