An matrix is an equitable matrix if all are positive, and for all .
Setting yields so diagonal elements of equitable matrices equal . Next, setting yields , so .
An example of an equitable matrix of order is
This example shows that equitable matrices exist for all .
The most general equitable matrix of orders and are
A matrix is equitable if and only if it can be expressed in the form
for real numbers with . (proof. (http://planetmath.org/ParameterizationOfEquitableMatrices))
An equitable matrix is completely determined by its first row. If , are known, then
If is an equitable matrix, then
where is the matrix exponential.
If is an equitable matrix and is a mapping, then
is an equitable matrix. In particular, striking the :th row and column in an equitable matrix yields a new equitable matrix.
- 1 H. Eves, Elementary Matrix Theory, Dover publications, 1980.