parameterization of equitable matrices
A matrix is equitable if and only if it can be expressed in the form
for real numbers with .
Assume that 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 being real numbers (which may be positive, negative or zero).
Since this comprises a system of linear equations for the quantities , 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 :
This simplifies to
In other words, all the diagonal entries are zero.
Consider the case when (but does not equal ).
By wat we have just shown, the right hand side of this equation equals zero. Hence, we have
In other words, the matrix of ’s is antisymmetric.
We may express any entry in terms of the entries :
We will conclude by noting that if, given any numbers with , but the remaining ’s arbitrary, we define
Hence, we obtain a solution of the equations
Moreover, by what we what we have seen, if we set , all solutions of these equations can be so described.
|Title||parameterization of equitable matrices|
|Date of creation||2013-03-22 14:58:36|
|Last modified on||2013-03-22 14:58:36|
|Last modified by||rspuzio (6075)|