On nonnegative matrices similar to positive matrices
Let \(\widetilde{\mathcal N}^n\) denote the set of complex \(n\)-tuples \((1, \lambda_2,\cdots,\lambda_n)\) such that there exists a nonnegative matrix with Perron root \(1\) and spectrum \(\{1, \lambda_2, \cdots, \lambda_n \}\). The authors prove that \(\widetilde{\mathcal N}^n\) is star-shaped with respect to \((1,0,\cdots,0)\) and that \((1, \lambda_2, \cdots, \lambda_n) \in \widetilde{\mathcal N}^n\) is on the boundary of \(\widetilde{\mathcal N}^n\) if and only if \(\{1, \lambda_2, \cdots, \lambda_n \}\) is not the spectrum of a positive matrix. Hence, they study the problem of determining the nonnegative pattern matrices \(P\) that are similar to positive pattern matrices. They obtain some partial results which help them in giving a complete solution to the case of \(3\times 3\) matrices.
scientific article; zbMATH DE number 1083226