On nonnegative matrices similar to positive matrices

Let Ñn denote the set of those (1, λ2, …, λn) such that there exists a nonnegative matrix with Perron root equal to one and spectrum {1, λ2, …, λn}. We prove that Ñn is star-shaped with respect to (1, 0, …, 0) and that (1, λ2, …, λn) Ñn is on the boundary of Ñn if and only if {1, λ2, …, λn} is not the spectrum of any positive matrix. As a consequence, attention is given to the problem of determining which nonnegative matrices are similar to positive ones. More precisely, we address the question of which pattern matrices P satisfy that any nonnegative matrix with pattern P is similar to a positive matrix. Some partial results are obtained (among them that any irreducible nonnegative matrix with a positive line is similar to a positive matrix), which allow us to give a complete solution to the case of 3-by-3 matrices.