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.