Characterization of a matrix with eigenvalues equal to one

Question

Consider an $m\times m$ non-negative matrix $A$ where elements of $A$ can take many different values e.g. they are functions of a variable z. Suppose $A$ is such that one of its eigenvalues is equal to one. Can we say anything about the properties of matrix $A$?

For example, a sufficient condition is that the sum of all columns to be one [plus irreducibility]. Under this condition, irrespective of the values of the elements of the matrix, one eigenvalue is always equal to one. My question: is there a simple necessary condition?

Answer 1

The $z$ is a red herring: what you can say about $A(z)$ is exactly what you can say about a single non-negative matrix whose eigenvalues are all equal to $1$.

Here's one thing you can say. Consider the directed graph $G$ corresponding to vertices $1, \ldots, n$ with an arc from $i$ to $j$ iff $A_{ij} > 0$.
If $S$ is a set of vertices such that

1. There is no arc from $S$ to $S^c$.
2. $S$ is strongly connected, i.e. for every $i, j \in S$ there is a chain of arcs from $i$ to $j$ and a chain of arcs from $j$ to $i$.

then $S$ has just one member.