The Case of Equality in Hopf’s Inequality

Abstract

Hopf’s inequality states that the subdominant eigenvalues $\lambda $ of a positive n-square matrix A satisfy \[ | \lambda |\leqq \frac{M - m}{M + m} \lambda _p \] where $\lambda _p $ is the Perron eigenvalue of A and M, m are, respectively, the maximum and minimum entries of A. A complete analysis of the case of equality in Hopf’s inequality is given. If A has an eigenvalue $\lambda $ which satisfies the case of equality, it is shown that $\lambda $ is real and the structure of the matrix A is determined.