Study on the Similarity between Nonnegative Irreducible Matrices and Positive Matrices

Abstract

The main objective of this study is to explore the problem of similarity between nonnegative irreducible matrices and positive matrices. By leveraging established lemmas and drawing insights from pertinent literature, this article provides a rigorous proof: When  is greater than 8, for any  nonnegative irreducible matrix , if it contains  zero elements and possesses mutually distinct diagonal elements, then  is similar to a positive matrix.