A characterization of trace zero nonnegative \(5\times 5\) matrices (Q1970444)

The determination of necessary and sufficient conditions for the existence of an entrywise nonnegative \(n \times n\) matrix \(A\) with spectrum \(\sigma\) is presented for the case \( n=5 \) with the trace equal to zero i. e. \(\lambda_1 +...+ \lambda_5 = 0\), where \(\lambda_i\) are complex numbers. This determination using the notion of \(n\)-cycle products is a part of the nonnegative inverse eigenvalue problem.