We generalize the Perron–Frobenius Theorem for nonnegative matrices to the class of nonnegative tensors.

1. J. Douglas, Jr. and T. M. Gallie, An approximate solution of an improper boundary value problem, Duke Math. J. vol. 26 (1959) pp. 339-347. 2. F. John, Numerical solution of the equation of heat conduction for preceding times, Ann. Mat. Pura Appl. ser. IV vol. 40 (1955) pp. 129-142. 3. C. Pucci, Sui problemi di Cauchy non "ben posti," Atti Accad. Naz. Lincei. Rend. Cl. Sei. Fis. Mat. Nat. vol...

By the use of Perron-Frobenius theory, simple proofs are given of the Fundamental Theorem of Demography and of a theorem of Cushing and Yicang on the net reproductive rate occurring in matrix models of population dynamics. The latter result, which is closely related to the Stein-Rosenberg theorem in numerical linear algebra, is further refined with some additional nonnegative matrix theory. Whe...

If A is a nonnegative matrix whose associated directed graph is strongly connected, the Perron-Frobenius theorem asserts that A has an eigenvector in the positive cone, (R). We associate a directed graph to any homogeneous, monotone function, f : (R) → (R), and show that if the graph is strongly connected then f has a (nonlinear) eigenvector in (R). Several results in the literature emerge as c...

This paper provides a simple proof for the Perron-Frobenius theorem concerned with positive matrices using a homotopy technique. By analyzing the behaviour of the eigenvalues of a family of positive matrices, we observe that the conclusions of Perron-Frobenius theorem will hold if it holds for the starting matrix of this family. Based on our observations, we develop a simple numerical technique...

A matrix is said to have the Perron-Frobenius property if its spectral radius is an eigenvalue with a corresponding nonnegative eigenvector. Matrices having this and similar properties are studied in this paper as generalizations of nonnegative matrices. Sets consisting of such generalized nonnegative matrices are studied and certain topological aspects such as connectedness and closure are pro...