Nonsingular H-matrices

H -matrix is an actual context of a broad category matrix, as we all know, including mathematics,physics,mechanics and engineering mathematics, the last of the many practical problems often attributed to one or a number of large scale sparse matrix linear algebraic equations to solve , and we often assume the coefficient matrix is H -matrix in the linear algebraic equations. At the same time, it plays an important role in cybernetics,electric system,economic science,elasticity mechanics and many other fields, however, the practical distinguishment of H -matrix is very difficult. So, it's very significant to judge if a matrix is H -matrix or not. In recent years, many domestic and overseas schlors have been woking on its properties and criterion, and have obtained lots of important results.The followings are the construction and main contents of this paper:The first part is introduction. In this part, we introduce the background H -matrix, and then introduce some questions are related with H -matrix.The second part is preliminaries. This part mainly make preparations for part four . In this part, some important definition and lemma are given, for example, M -matrix,H -matrix,diagonally dominant matrix and the lemma how to judgue a matrix is H -matrix.The third part is the conclusions have been proposed. In this part, we mainly intro- duce the criterion of H -matrix proposed by predecessors, such as document [1] to [6] give some sufficient condition to judge a matrix is H -matrix.The fourth part is the main conclusions. In this part, according to the definition of H -matrix , from the element of matrix itself, we give some practical criterion on the basis of our predecessors. The main method used is from the element of the matrix itself, structure some different tridiagonal matrix X , and combinate of inequality-reduction techniques So that AX is stiictly diagonal dominant matrix. Based on this, by H - matrix to determine an approach also gives the matrix, containing non-zero elements matrix to be an H -matrix .Numerical examples is the last part. In this part, we give some matrice, then use criterion is given in part four to verify the matrix is an H - matrix, and give the superiority with the text [5]and [6] in a comparison.