Journal of Applied Mathematics

Volume 2013 (2013), Article ID 676978, 6 pages

http://dx.doi.org/10.1155/2013/676978

Research Article

On the Positive Definite Solutions of a Nonlinear Matrix Equation

Panpan Liu,¹ Shugong Zhang,¹ and Qingchun Li²

¹School of Mathematics, Key Laboratory of Symbolic Computation and Knowledge Engineering (Ministry of Education), Jilin University, Changchun 130012, China

²Department of Mathematics, Beihua University, Jilin 132013, China

Received 28 November 2012; Revised 19 April 2013; Accepted 21 April 2013

Academic Editor: J. Biazar

Copyright © 2013 Panpan Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The positive definite solutions of the nonlinear matrix equation are discussed. A necessary and sufficient condition for the existence of positive definite solutions for this equation is derived. Then, the uniqueness of the Hermitian positive definite solution is studied based on an iterative method proposed in this paper. Lastly the perturbation analysis for this equation is discussed.

1. Introduction

Denote the set of all positive definite matrices by . In this paper, we consider the matrix equation

where is nonsingular, is a Hermitian positive definite matrix, is a positive real number, is a continuous map from   into , and is either monotone (meaning that implies that ) or antimonotone (meaning that implies that ).

Nonlinear matrix equation of the form (1) often arises in dynamic programming, control theory, stochastic filtering, statistics, and so on. In recent years, many authors have been much interested in studying this class of matrix equations [1–10].

Equation (1) has been investigated in some special cases. For the case , Ran and Reurings [1] derived some sufficient conditions for the existence and uniqueness of a positive definite solution of (1). In addition, an iterative method for obtaining Hermitian positive definite solutions of (1) with is proposed by Yueting [10]. Liu and Gao [9] proved the existence of the symmetric positive definite solutions of (1) with and . Many other authors investigated (1) for particular choices of the map [2–4, 6–8].

In Section 2, we will derive a necessary and sufficient condition for the existence of positive definite solutions of (1). In Section 3, we will propose an iterative method and investigate the uniqueness of the Hermitian positive definite solution. Finally, in Section 4, we will discuss the perturbation analysis of (1).

The following notations are used throughout this paper. For a positive definite matrix , and stand for the maximal and minimal eigenvalues of matrix , respectively. is the conjugate transpose of the matrix , and is the inversion of . denotes the spectral norm of . denotes that is a positive definite (semidefinite) matrix, and means that . The notation denotes the line segment joining and ; that is,

2. On the Positive Definite Solutions of (1)

In this section, we will derive a necessary and sufficient condition for the existence of positive definite solutions of (1).

Lemma 1 (see [8]). Assume that ; if , then , and if , then .

If there is unique Hermitian positive definite matrix , such that , then we denote that .

Theorem 2. Assume that is a continuous map from into , where denotes the set of all Hermitian positive definite matrices. Then, (1) has a Hermitian positive definite solution if and only if there is a nonsingular matrix , such that , and , where

In this case, (1) has a Hermitian positive definite solution .

Proof. If is a Hermitian positive definite solution of (1), then there is unique Hermitian positive definite matrix , such that (see [11]). So, , and therefore there is unique Hermitian positive definite matrix , such that . Substituting into (1) gives

Then, we have

is Hermitian positive definite, and so

that is

Let . Then, , and by (7) we know that .

Conversely, if , let . Then,

So, is a Hermitian positive definite solution of (1).

If , then the restriction in Theorem 2 can be omitted.

3. Iterative Method

In order to discuss an iterative method for solving (1), we assume that for a given matrix , the equation always has a positive definite solution and its solution is easy to obtain. We are interested in the inverse iteration, and consider the following iterative method:

In this section, we assume that , , in (1) satisfy .

Theorem 3. Suppose that exists and that is antimonotone. Let . Equation (1) has a positive definite solution in the interval if and only if there is a number , such that for all . Moreover, in this case, the iteration (9) with converges to the smallest positive definite solution of (1).

Proof. Since exists and is anti-monotone, then is also anti-monotone. Assume that there is a number as in the theorem. Since , one has

Furthermore, we get

Now, if , we have

Then, the sequence is a monotonically nondecreasing sequence and bounded above by some positive definite matrix . Consequently, the sequence converges to a positive definite matrix , which is a solution of (1); that is,

Conversely, let (1) have a positive definite solution , and let be the largest eigenvalue of . In order to prove that for every generated from (9), we will prove that . Consider

Assume that , for some fixed , and then we have

So, . Then, the theorem is proved.

Lemma 4 (see [1]). Let be differentiable at any point of . Then,

for all .

Lemma 5. Suppose that exists and that is anti-monotone. If (1) with has a Hermitian positive definite solution , then .

Proof. Since exists and is anti-monotone, then is also anti-monotone. Assume that (1) has a Hermitian positive definite solution . Since maps into , we have and . Therefore, . By Lemma 1 we know that . On the other hand, we have , . Then, we obtain because is anti-monotone.

Theorem 6. Suppose that exists and that is anti-monotone, and suppose that , are differentiable at any point of and , respectively. Let

and .

(i)If (1) with has a Hermitian positive definite solution and , then is the unique solution of (1).(ii)Assume that there is a closed set satisfying that and ; if , then (1) with has a unique solution in . Furthermore, one considers the iterative method (9) with . The sequence in (9) converges to the unique solution ; moreover,

Proof. (i) Assume that and are two different Hermitian positive definite solutions of (1), and by Lemma 5, ; then, . Let , and

From Lemma 4,

Let

Then,

By we have

which is a contradiction; so, . That is, is the unique solution of (1).

(ii) Let , . Then, ; from Lemma 4,

The interval is a complete metric space because it is a closed subset of . And ; so, is a contraction on . Then, it follows from contractive mapping principle that the map has a unique fixed point in . Furthermore, the sequence in (9) converges to the unique solution of (1); moreover,

4. Perturbation Analysis

Let

be the perturbation equation of (1). Let , be nonsingular matrices and , be positive definite, and . Suppose that exists and is anti-monotone.

Lemma 7 (see [12]). If , the operators , satisfy and for some positive number ; then,

Theorem 8. Let , be the positive definite solutions of (1) and its perturbation equation (26), respectively. Map is differentiable at any point of with

Let

If , one has

where , .

Proof. , are the positive definite solutions of (1) and (26), respectively. Let , . From Lemma 4,

By Lemma 5,

Let

Then, we have

Notice that

Let , . Then, , . By Lemma 1, , since . And from Lemma 7,

Therefore,

That is,

If , by we have

where

Acknowledgments

The authors wish to thank Professor J. Biazar and the reviewers for providing valuable comments and suggestions which improve this paper. This work is supported by NSFC (Grant no. 11171133).

References

A. C. M. Ran and M. C. B. Reurings, “On the nonlinear matrix equation : solutions and perturbation theory,” Linear Algebra and Its Applications, vol. 346, pp. 15–26, 2002.

I. G. Ivanov, “On positive definite solutions of the family of matrix equations ,” Journal of Computational and Applied Mathematics, vol. 193, no. 1, pp. 277–301, 2006.

J. F. Wang, Y. H. Zhang, and B. R. Zhu, “The Hermitian positive definite solutions of the matrix equation ,” Mathematica Numerica Sinica, vol. 26, no. 1, pp. 61–72, 2004 (Chinese).

S. M. El-Sayed and A. M. Al-Dbiban, “On positive definite solutions of the nonlinear matrix equation ,” Applied Mathematics and Computation, vol. 151, no. 2, pp. 533–541, 2004.

S. M. El-Sayed and A. C. M. Ran, “On an iteration method for solving a class of nonlinear matrix equations,” SIAM Journal on Matrix Analysis and Applications, vol. 23, no. 3, pp. 632–645, 2001/02.

S. M. El-Sayed and M. G. Petkov, “Iterative methods for nonlinear matrix equations ,” Linear Algebra and Its Applications, vol. 403, pp. 45–52, 2005.

V. I. Hasanov, “Positive definite solutions of the matrix equations ,” Linear Algebra and Its Applications, vol. 404, pp. 166–182, 2005.

X. F. Duan and A. P. Liao, “The Hermitian positive definite solution of the matrix equation and its perturbation analysis,” Numerical Mathematics, vol. 30, no. 3, pp. 280–288, 2008 (Chinese).

X.-G. Liu and H. Gao, “On the positive definite solutions of the matrix equations ,” Linear Algebra and Its Applications, vol. 368, pp. 83–97, 2003.

Y. Yueting, “The iterative method for solving nonlinear matrix equation ,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 46–53, 2007.

G. W. Stewart and J. G. Sun, Matrix Perturbation Theory, Computer Science and Scientific Computing, Academic Press, Boston, Mass, USA, 1990.

R. Bhatia, Matrix Analysis, vol. 169 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1997.