Jordan Structures of Totally Nonnegative Matrices An $n \times n$ matrix is said to be totally nonnegative if every minor of $A$ is nonnegative. In this paper we completely characterize all possible Jordan canonical forms of irreducible totally nonnegative matrices. Our approach is mostly combinatorial and is based on the study of weighted planar diagrams associated with totally nonnegative matrices. Keywords:totally nonnegative matrices, planar diagrams,, principal rank, Jordan canonical formCategories:15A21, 15A48, 05C38