Spectral structures of irreducible totally nonnegative matrices

An n-by-n matrix is called totally nonnegative if every minor of A is nonne gative. The problem of interest is to characterize all possible Jordan cano nical forms ( Jordan structures) of irreducible totally nonnegative matrice s. We show that the positive eigenvalues of such matrices have algebraic mu ltiplicity one, and also demonstrate key relationships between the number a nd sizes of the Jordan blocks corresponding to zero. These notions yield a complete description of all Jordan forms through n = 7, as well as numerous general results. We also de ne a notion of principal rank and employ this idea throughout.