On nonnegative matrices similar to positive matrices

Let (N) over tilde(n) denote the set of those (1, lambda(2),..., lambda(n)) is an element of C-n such that there eldsts a nonnegative matrix with Perron root equal to one and spectrum {1, lambda(2),..., lambda(n)}. We prove that (N) over tilde(n) is star-shaped with respect to (1, 0,..., 0) and that (1, lambda(2),..., lambda(n)) is an element of (N) over tilde(n) is on the boundary of (N) over tilde(n) if and only if {1, lambda(2),..., lambda(n)} 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. (C) 1997 Elsevier Science Inc.