Upper bounds for eigenvalues of symmetric substochastic matrices and applications

This paper presents the upper bounds for eigenvalues of the symmetric substochastic matrices, which are related to the measure of irreducibility of matrices. Then three applications in different fields are presented to indicate further applications. In the first one, the bounds for eigenvalues are used to prove that a class of iteration systems could reach a consensus under certain conditions. In the second one, they are used to prove the flocking behaviors of the Cucker–Smale model under rooted leadership. In the last one, a new approach to estimate the upper bounds for eigenvalues of nonnegative matrices is provided.