As I understand, Perron Frobenius theorem asserts only in one direction, i.e. if Matrix A is positive then there is a perron eigenvalue, eigenvector etc.
What I wanted to know is what are the conditions on the eigenvalues/eigenvectors in order to get a positive matrix?
[Wikipedia] (http://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem#Positive_matrices/) point 4 states that all eigenvectors except the eigenvector corresponding to the perron eigenvalue must have at least one component negative or complex. Is that necessary/sufficient to get a positive matrix?
In other words, suppose I wanted to construct a matrix by choosing its eigenvalues and eigenvectors. If I adhered to the following points, would I always get a strictly positive (or non-negative) matrix? If not, what additional conditions would be required?
- One perron eigenvalue (all other eigenvalues strictly lesser in magnitude) with a one dimensional eigenspace.
- A strictly positive eigenvector corresponding to the perron eigenvalue
- All other eigenvectors have at least one component negative or complex
If it makes it any easier, you can assume that we want to construct an irreducible stochastic matrix i.e. perron eigenvalue 1 and corresponding eigenvector of all 1s