The paper gives new observations on irreducible nonnegative realizations of realizable spectra and characterizes the trace-zero case via graph theory, plus related positive realizations.
Brauer, Limits for the characteristic roots of a matrix, IV: Applications to stochastic ma- trices,Duke Math
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The INIEP: Irreducible and Positive Realizations
The paper gives new observations on irreducible nonnegative realizations of realizable spectra and characterizes the trace-zero case via graph theory, plus related positive realizations.