M-matrix Subsets Closed Under Addition And Some Applications

It is well known that nonsingular M-matrices, like symmetric positive definite matrices, have a lot of important applications. Also, the two classes of matrices have a lot of similar properties. For example, their principal submatrices have the inheritance property, all of their principal minors are positive, and all of real parts of their eigenvalues are positive, etc.. The sum of two symmetric definite matrix is still symmetric definite, but how about the sum of two noningular M-matrix? The answer is negative. In this article, we present some noningular M-matrix subsets which are closed under addition: subpositive definite M-matrix, upper triangular M-matrix, diagonally dominant M-matrix with nonzero chain, k-circular M-matrix and so on. At last we extend the conclusion to H-matrix and present some inverse M-matrix subsets which are closed under the parallel sum. We also discuss in what case spectrum radius of sum of two nonnegative matrices is no bigger than sum of spectrum radius of the two nonnegative matrices.