An inequality for the spectra of nonnegative matrices ∗

Let A be a n×n entrywise nonnegative matrix and let sk:=trace(Ak),k=1,2,3. It is shown that if n>1 then is nonnegative. The result is used to show that if is the spectrum of a nonnegative matrix where λ2 is nonreal and λ1=max(|λj|,j=1,…,n) then need not be realizable for all t>0 even when Re(λ2)⩾0.