Abstract:The rank and the number of non-zero eigenvalues of a matrix are two important invariants and the relation between these two values is a basic problem in the linear algebra. Some authors have described the necessary and sufficient conditions for that the rank and the number of non-zero eigenvalues are equal or have a gap of n-1. On the other side, the index of matrix is another important invariant, which, roughly speaking, is the maximal size of the zero eigenvalues in the canonical form of a complex matrix. Based on the existing research results, the relation of the gap between the rank and the number of non-zero eigenvalues with the index of matrix was investigated, and the necessary and sufficient conditions for these invariants were obtained, which is a generalization of some known results.