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In this paper, according to the theory of two classes of $ \alpha $-diagonally dominant matrices, the row index set of the matrix is divided properly, and then some positive diagonal matrices are constructed. Furthermore, some new criteria for nonsingular $ H $-matrix are obtained. Finally, numerical examples are given to illustrate the effectiveness of the proposed criteria.

This paper focuses on the state feedback stabilization problem for a class of discrete-time singular Markov jump systems with repeated scalar nonlinearities. First, on the basis of the implicit function theorem and the diagonally dominant Lyapunov approach, a sufficient condition is obtained, which ensures the regularity, causality, uniqueness of solution in the neighbourhood of the origin, and stochastic stability for the system under consideration. Moreover, by employing some lemmas and matrix inequalities, the sufficient condition is changed into a set of linear matrix inequalities. Then, the procedures of designing the state feedback controller are given. Eventually, three examples are presented to show the validness of the proposed approach.