#### Abstract

Comparison theorems between the spectral radii of different matrices are useful tools for judging the efficiency of preconditioners. In this paper, some comparison theorems for the spectral radii of matrices arising from proper splittings of different semimonotone matrices are presented.

#### 1. Introduction and Preliminaries

Let be the null matrix with suitable size. The notation denotes that all entries of matrix are nonnegative (positive), and in this case matrix is called nonnegative (positive). For two real matrices, , means that . The same notation is valid for vectors. A real rectangular matrix is said to be semimonotone if [1]; here is the Moore-Penrose inverse of , that is, the unique matrix which satisfies the Moore-Penrose equation , , , and ( denotes the transpose of ); see [2, 3].

Real rectangular linear system of the form where is a real matrix and is a real -vector, appears in many areas of mathematics. For example, finite difference discretization of partial differential equations with Neumann boundary conditions. Iterative methods for solving (1) can be formulated by the decomposition of as [4], and the approximation solution of (1) is generated by

The decomposition is called a proper splitting if and [4], where and are the range and kernel of , respectively. Let be the spectral radius of the real square matrix ; then, for the proper splitting , the iterative scheme (2) converges to the minimal norm least square solution of (1) for any initial vector if and only if [4, Corollary 1]. In this case, we say that the proper splitting is a convergent splitting. Moreover, the fact that is a proper splitting, as is a proper splitting, implies that and is invertible, so we have [1, Theorem 2.2] and . The next lemma shows the relation between the eigenvalues of and .

Lemma 1 (see [1, Lemma 2.6]). *Let be a proper splitting of real matrix . Let , and , be the eigenvalues of and , respectively. Then for every , we have . Also, for every , there exists such that and, for every , there exists such that .*

For nonnegative matrix, there is a well-known result which is shown next.

Lemma 2 (see [5, Theorem 2.21]). *Let be matrices; if , then .*

Using the notation of nonnegative matrix, the proper regular and proper weak regular splittings, which are the natural extensions of the regular and weak regular splittings of a real square matrix [5, 6], are defined as follows.

*Definition 3. *For a real matrix , the splitting is called (1)proper regular splitting if it is a proper splitting such that and [7, Definition 1], [8, Definition 1.2];(2)proper weak regular splitting of first type if it is a proper splitting such that and ; proper weak regular splitting of second type if it is a proper splitting such that and [7, Definition 1], [8, Definition 1.2].

It should be remarked that Jena et al. [8] only considered proper weak regular splitting of first type; they name it as proper weak regular splitting. The existence of the proper splitting is discussed in [4]; there is an example in [4] to show how to construct such splitting.

Let be a proper regular splitting of ; Berman and Plemmons in [4] showed that if and only if . Other convergence results of proper regular and/or weak regular splitting can be found in [8, 9]. Comparison theorems between the spectral radii of matrices are useful tools for analyzing the rate of convergence of iterative methods or for judging the efficiency of preconditioners [8, 10–12]. There is also a connection to population dynamics [11]. Some comparison theorems of proper splittings of a semimonotone matrix are established recently in [8, 13].

Our basic purpose here is to give a new convergence theorem for proper weak regular splitting of a semimonotone matrix and to derive the comparison theorems of proper regular and proper weak regular splittings of different semimonotone matrices. The condition of new convergence theorem is weaker than that in [8], and the comparison results generalized the corresponding results in [5, 8, 11]. The comparison results can be further used for judging the efficiency of the preconditioners.

#### 2. Main Results

Recall that the proper regular splitting of a semimonotone matrix is a convergent splitting [4, 8]. For proper weak regular splitting of a semimonotone matrix, we have the following convergence theorem.

Theorem 4. *Let be a proper weak regular splitting (of any type) of real matrix . If and , then
*

*Proof. *Note that ; the proof is essentially analogous to that in [8]. We omit it here.

*Remark 5. *Jena et al. [8] concluded that, for a proper weak regular splitting of real matrix , the convergence conditions are and , so the condition of Theorem 4 is weaker than that in [8]. To see this, let
Then is a semimonotone matrix and is a proper weak regular splitting of . It is easy to see that and , but does not hold.

Let and be two semimonotone matrices and let and be proper splittings of and , respectively. In what follows, we will present the comparison results between and . The comparison theorems for proper regular splittings are given first.

Theorem 6. *Let and be two semimonotone matrices and let and be proper regular splittings of and , respectively. If and , then
*

*Proof. *As and are semimonotone matrices and and are proper regular splittings, it follows from [4] that for . Thus all we need to show is .

For , note that the matrices are nonnegative; Perron-Frobenius theorem (cf. [5]) states that the spectral radius of is an eigenvalue corresponding to a nonnegative eigenvector; then from Lemma 1, is an eigenvalue of ; hence, . Again, by Perron-Frobenius theorem, implies existence of a nonnegative vector such that . Then
implies is an eigenvalue of ; hence, ; that is, . Therefore, we have
Note that ; then, and lead to , and Lemma 2 yields . Let ; then, is a strictly increasing function for . Hence the inequality holds.

*Remark 7. *The assumptions and of Theorem 4 can be weakened as .

For different proper regular splittings of one semimonotone matrix , the following corollary is obtained.

Corollary 8 (see [8, Theorem 3.2]). *Let be a semimonotone matrix and let be two proper regular splittings of . If , then
*

When we consider the monotone matrices, we have the following corollaries directly.

Corollary 9 (see [11, Theorem 4.2]). *Let and be two monotone matrices and let and be regular splittings of and , respectively. If and , then
*

Corollary 10. *Let and be two monotone matrices and let and be regular splittings of and , respectively. If , then
*

Corollary 11 (see [5, Theorem 3.32]). *Let be a monotone matrix and let be two regular splittings of . If , then
*

Next the comparison results for proper weak regular splittings are given.

Theorem 12. *Let and be two semimonotone matrices and let and be proper weak regular splittings of the same types of and , respectively. If , then
*

*Proof. *Note that ; from Theorem 4 we have . Analogous to the proof of Theorem 6, the desired comparison results are obtained.

Theorem 13. *Let and be two semimonotone matrices and let and be proper weak regular splittings of different types of and , respectively. Assume that and . If , then
*

*Proof. *Since is a proper weak regular splitting of semimonotone matrix and , it follows from Theorem 4 that . Hence, it suffices to show that .

Assume first that is of second type and is of first type. Note that the splittings and are proper splittings; then, , , for , and and (see, e.g., [3, Exercise 1.3(2)]). Using we obtain
For and , by Perron-Frobenius theorem (cf. [5]), there exist two nonzero vectors and such that
Thus
By assumption we obtain
Therefore
The case that is of first type and is of second type can be proved in a similar way.

The proof is completed.

When considering the monotone matrices, the condition for the convergence of weak regular splitting (weak nonnegative splitting in [11]) is not necessary. Hence we have the following corollary.

Corollary 14. *Let and be two monotone matrices and let and be weak regular splittings of different types of and , respectively. Assume that . If , then
*

#### 3. Conclusion

In this paper, a new convergence theorem for proper weak regular splitting of a semimonotone matrix and two comparison theorems for proper weak regular and proper weak regular splittings of different semimonotone matrices are given. The obtained results are improved and/or generalized the previous results. Applying the comparison results to judge the efficiency of the preconditioners for rectangular linear system needs further study.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors would like to thank two anonymous referees and Dr. Zhiping Xiong of Wuyi University for their valuable comments and suggestions, which improve the presentation of this paper. This work was supported by the National Natural Science Foundation of China (Grant no. 11301290) and the Youth Research Ability Project of Northwest Normal University (Grant no. NWNU-LKQN-13-15).