NUMERICAL ALGEBRA, doi:10.3934/naco.2016002 CONTROL AND OPTIMIZATION Volume 6, Number 2, June 2016 pp. 103–113 INDEX-PROPER NONNEGATIVE SPLITTINGS OF MATRICES Chinmay Kumar Giri Department of Mathematics National Institute of Technology Raipur Raipur - 492 010, India (Communicated by Hua Dai) Abstract. The theory of splitting is a useful tool for finding solution of a system of linear equations. Many woks are going on for singular system of linear equations. In this article, we have introduced a new splitting called index- proper nonnegative splitting for singular square matrices. Several convergence and comparison results are also established. We then apply the same theory to double splitting. 1. Introduction. Iterative methods for solving the system of linear equations Ax = b, (1) where A is a real square n×n matrix and b is a real n-vector, are related to splittings of A (a splitting is an expression of the form A = U −V , where U and V are matrices of same order as in A). A splitting A = U − V is called an index-proper splitting ([6]) of A ∈ R n×n if R(A k )= R(U k ) and N (A k )= N (U k ), where k = ind(A) (see next section for its definition), and R(A) and N (A) stand for the range of A and the kernel of A. It reduces to index splitting ([14]) if ind(U )=1. When k = 1, then an index-proper splitting becomes a proper splitting ([4]). The asymptotic behavior of the iterative sequences: x i+1 = U D Vx i + U D b, i =0, 1, 2,... and Y j+1 = U D VY j + U D ,j =0, 1, 2,..., where U D is the Drazin inverse of U , is governed by the spectral radius of the iteration matrix U D V (see next section for the definition of Drazin inverse). For an index-proper splitting, the spectral radius of U D V is strictly less than 1 if and only if the above schemes converge to A D b and A D , respectively to the system Ax = b. More on index-proper splitting can be found in the recent articles [6, 7]. The aim of this paper is to study the theory of nonnegative splitting 1 for square singular matrices using Drazin inverse. When two splittings of A are given, it is of interest to compare the spectral radii of the corresponding iteration matrices. The comparison of asymptotic rates of con- vergence of the iteration matrices induced by two index-proper splittings of a given 2010 Mathematics Subject Classification. 15A09, 65F15, 65F20. Key words and phrases. Drazin inverse; Group inverse; Non-negativity; Index-proper splitting; Convergence theorem; Comparison theorem. 1 A splitting U - V is called nonnegative ([12]) if U -1 exists and U -1 V ≥ 0. 103