INDEX-PROPER NONNEGATIVE SPLITTINGS OF MATRICES

. 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.


Introduction. Iterative methods for solving the system of linear equations
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 V x i + U D b, i = 0, 1, 2, . . . and Y j+1 = U D V Y 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 convergence of the iteration matrices induced by two index-proper splittings of a given 104 CHINMAY KUMAR GIRI matrix, has been studied in the recent articles by Jena and Mishra, [6] and Jena and Pani, [7]. Jena and Mishra,[6] also obtained many results for nonnegativity of the Drazin inverse using different matrix splittings. Applications of Drazin inverse lie in many areas such as singular differential and difference equations, Markov chain, cryptography, iterative methods, multi-body dynamics and optimal control. Therefore the computation of the Drazin inverse and its properties have been an area of active research. Here, only a few articles on the Drazin inverse are mentioned, but there is a vast amount of literature on it. (See the references [2,3,6,7,14] and the references cited therein.) In this article, we first introduce a new splitting called index-proper nonnegative splitting (see Section 3 for the definition) for square singular matrices by extending the notion of nonnegative splitting for square nonsingular matrices using the notion of Drazin inverse. We then study the convergence of this splitting and prove comparison results for different iterative schemes arising out of this splitting. At last, we apply to theory of double index-proper splitting.
The organization of this paper is as follows. In Section 2, we list all relevant definitions, notation and some earlier results which we use throughout the paper. The main results are given in Section 3 and Section 4. Section 3 introduces the generalization of nonnegative splitting to square singular matrices, and then discusses convergence and comparison theorems for this splitting. In Section 4, we propose the notion of double index-proper nonnegative splitting for real n×n matrices. Then convergence and comparison results for double index-proper nonnegative splitting are established.

2.
Preliminaries. Throughout this article, we will deal with R n equipped with its standard cone R n + , and all our matrices are real square matrices of order n unless stated otherwise. We denote the transpose, the null space and the range space of A by A T , N (A) and R(A), respectively. A is said to be nonnegative (i.e., A ≥ 0) if all the entries of A are nonnegative, and B ≥ C for matrices B and C, if B −C ≥ 0. We also use these notation and nomenclature for vectors. Let L, M be complementary subspaces of R n . Then P L,M stands for the projection of The index of A is the least nonnegative integer k such that rank(A k+1 )=rank(A k ), and we denote it by ind (A). Then ind (A) = k if and only if R(A k ) ⊕ N (A k ) = R n . Also, for l ≥ k, R(A l ) = R(A k ) and N (A l ) = N (A k ). The Drazin inverse of a matrix A ∈ R n×n is the unique solution X ∈ R n×n satisfying the equations: A k = A k XA, X = XAX and AX = XA, where k is the index of A. It is denoted by A D . When k = 1, then the Drazin inverse is said to be the group inverse and is denoted by A # . While Drazin inverse exists for all matrices, the group inverse does not. It exists if and only if ind (A) = 1. If A is nonsingular, then of course, we have We list certain results to be used in the sequel. The next two theorems deal with nonnegativity and spectral radius, and the first one is known as Perron−Frobenius theorem which states that: Theorem 2.1. (Theorem 2.20, [13]) Let A ≥ 0. Then A has a nonnegative real eigenvalue equal to its spectral radius.  The next result is finite dimensional version of the corresponding result which also holds in Banach spaces.
The first result given below shows that U D V and A D V have the same eigenvectors while the next one provides a relation between their eigenvalues.  and for every j, there exists i such that λ j = µi 1−µi . We conclude this section with the following two theorems.
So, the following results are: 3. Index-proper nonnegative splitting. In this section, we first introduce the definition of an index-proper nonnegative splitting. After that some convergence and comparison theorems are proved under a few different sufficient conditions. Very recently, Baliarsingh and Jena [2] have introduced index-proper regular and index-proper weak regular splittings for singular square matrices. Jena [5] again studied the same theory.
We now introduce another splitting which covers a larger class of matrices in comparison to the above two splittings.
When k = 1, the above definition coincides with the definition of proper Gnonnegative splitting (Definition 5.1, [10]). So index-proper nonnegative splitting is an extension of proper G-nonnegative splitting. While the theory of proper regular, proper weak regular and proper nonnegative splitting for rectangular matrices are recently studied in the articles [8] and [10].
The example given below does not has an index-proper regular or an index proper weak regular splitting, but has an index-proper nonnegative splitting. Not only that it also does not have Proper G-nonnegative splitting.
is an index-proper nonnegative splitting but not an index-proper regular and weak regular splitting as V ≥ 0 and U D ≥ 0, respectively.
Note that with a different U , A may have an an index-proper regular or an index proper weak regular splitting. Thus the class of index-proper nonnegative splitting is larger than the class of index-proper regular and weak regular splittings. We next present a convergence result for index-proper nonnegative splitting under a sufficient condition A D U ≥ 0.
. Now we discuss some more properties of index-proper nonnegative splitting.
Let λ and µ be any nonnegative eigenvalues of A D V and U D V , respectively. Let f (λ) = λ 1+λ , λ ≥ 0. Then f is a strictly increasing function. Then, by Lemma 2.6, µ = λ 1+λ . So, µ attains its maximum when λ is maximum. But λ is maximum when λ = ρ(A D V ). As a result, the maximum value of µ is ρ( The theorem given below generalizes Lemma 2.6, [12] to square singular matrix case. Mishra [10] also extended the same lemma to rectangular case using the Moore-Penrose inverse. But we have used the Drazin inverse here. The Drazin inverse case does not hold in general. It is true under the assumption of two extra conditions and is presented next. It is well known that the comparison theorems between the spectral radii of matrices are useful in analysis of rate of convergence of iterative methods induced by the splittings and iteration matrices. An accepted rule for preferring one iteration scheme to another is to choose the scheme having the smaller spectral radius of U D V . Many authors such as Mishra [10], Jena et al. [8], Jena and Mishra [6], etc. have introduced various comparison results for different splittings of semimonotone (i.e. A † ≥ 0) and Drazin monotone matrices. Here we discuss the case of indexproper nonnegative splitting.
Next two comparison results are for index-proper nonnegative splittings of a Drazin monotone matrix.

Proof. By Lemma 3.4, the conditions
4. Double index-proper nonnegative splittings. Motivated by the idea of Jena et al. [8] we now introduce the double index-proper splitting A = P − R − S of A to Ax = b which leads to the following iterative scheme spanned by three iterates: Then The iteration scheme (3) is convergent if ρ(W ) < 1 and then A = P − R − S is called a convergent double splitting.
Setting P = U and R − S = V in Definition 4.1 and k = 1, we obtain proper nonnegative splitting (Definition 3.1, [10]) for square matrices. We first prove a convergence result which relates convergence of single and double splitting. Note that Mishra, [10] also studied a similar result for rectangular matrices while our result discusses about only square singular matrices.    Proof. The proof of this theorem is same as above. By taking U = P and V = R+S, we get A = U − V is an index-proper nonnegative splitting. Then, by (a) ⇒ (b) of Theorem 3.5, we get ρ(P D (R + S)) = ρ(U D V ) < 1. Again, by Lemma 2.3, it now follows that ρ(W ) < 1.
A comparison result is shown next which discusses comparison of two different system of linear equations having two different double index-proper nonnegative splitting under a few sufficient conditions. The proof is analogous to Theorem 4.9, [10].
Proof. By Theorem 4.3, we have ρ(W i ) < 1 for i = 1, 2. If ρ(W 1 ) = 0, then our claim holds trivially. Suppose that ρ(W 1 ) = 0. Since A 1 and A 2 possesses double index-proper nonnegative splitting, so W 1 ≥ 0 and W 2 ≥ 0. Now, applying Lemma 2.1 to W 1 , we get W 1 x = ρ(W 1 )x i.e., The next example shows that the converse of Theorem 4.5 is not true. A 2 = P 2 − R 2 − S 2 are two double index-proper nonnegative splittings. We then have 0.5 = ρ(W 1 ) ≤ ρ(W 2 ) = 0.5 < 1, but P D 1 A 1 ≥ P D 2 A 2 and P D 1 R 1 ≥ P D 2 R 2 . Let A = P 1 − R 1 − S 1 = P 2 − R 2 − S 2 be two double index-proper nonnegative splitting of A. Then, we have W 1 = P D 1 R 1 P D 1 S 1 I O ≥ 0 and W 2 = P D 2 R 2 P D 2 S 2 I O ≥ 0. We finally conclude with a comparison theorem for a single system of equation where the coefficient matrix A has two different double index-proper nonnegative splitting.