ELA, Volume 9, pp. 270-275, September 2002, abstract.  

Construction of Trace Zero Symmetric Stochastic
Matrices for the Inverse Eigenvalue Problem

Robert Reams

In the special case of where the spectrum
   sigma={lambda_1,lambda_2,lambda_3,0,0,...,0} 
has at most three nonzero eigenvalues lambda_1, 
lambda_2, lambda_3 with 
   lambda_1 >= 0 >= lambda_2 >= lambda_3, 
and 
   lambda_1 + lambda_2 + lambda_3 = 0, 
the inverse eigenvalue problem for symmetric 
stochastic nxn matrices is solved. Constructions 
are provided for the appropriate matrices where 
they are readily available. It is shown that when 
n is odd it is not possible to realize the spectrum 
sigma with an nxn symmetric stochastic matrix 
when lambda_3 is nonzero and
     3/(2n-3) > lambda_2/lambda_3 >= 0, 
and it is shown that this bound is best possible.