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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 216035, 4 pages
Existence and Uniqueness of the Positive Definite Solution for the Matrix Equation
Department of Mathematics, Heze University, Heze, Shandong 274015, China
Received 15 May 2013; Revised 23 June 2013; Accepted 4 July 2013
Academic Editor: Vejdi I. Hasanov
Copyright © 2013 Dongjie Gao. 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.
We consider the nonlinear matrix equation , where is positive definite, is positive semidefinite, and is the block diagonal matrix defined by . We prove that the equation has a unique positive definite solution via variable replacement and fixed point theorem. The basic fixed point iteration for the equation is given.
We consider the matrix equation where is an positive definite matrix, is an positive semidefinite matrix, is arbitrary matrix, and is the block diagonal matrix defined by in which is an matrix. This matrix equation is connected with certain interpolation problem (see ).
When , we write as the block matrix where are matrices. Then (1) is equal to , which has been investigated by many authors (see [2–9]). However, are few little theoretical results when . Ran and Reurings  have proved that (1) has a unique positive definite solution. Based on this, Sun  presented some perturbation results for the unique solution.
Fixed point theorem is often used to discuss the existence and uniqueness of the solution. However, for (1), it is difficult to prove the uniqueness of the solution directly by using fixed point theorems. In this paper, we turn (1) into its equivalent equation via variable replacement. Then we consider its equivalent equation by using fixed point theorem. This provides a new proof for the existence and uniqueness of the positive definite solution. And this method is shown to be much easier than the way of . In addition, the basic fixed point iteration for the equation is given.
In this paper we use to denote complex matrices, to denote positive definite matrices, and to denote positive semidefinite matrices. For , we write if is positive semidefinite (definite). denotes the conjugate transpose of a matrix . Let , , , and .
2. The Case of
In this section, we discuss (1) with . Let be the set defined by We are interested in positive definite solutions of (1) in this set. A matrix is a solution of (1) if and only if it is a fixed point of the map defined by Let . For , we know that is a positive definite matrix. Then (1) turns into Let . Then (1) eventually becomes
Theorem 1. Equation (6) has a positive semidefinite solution for any .
Proof. A matrix is a solution of (6) if and only if it is a fixed point of the map defined by . Note that maps into itself, because is order reversing. Hence it has a fixed point in the . That is to says (6) has a positive semidefinite solution.
Theorem 2. If is positive semidefinite solution of (6), then .
Proof. By , and , we know that and ; then . By , we know that . That is, .
For proving the uniqueness, we first verify the following lemma.
Lemma 3. Let . For any and , one has
Proof. By Theorem 2, we know for any . Then for any . Let , and . For any and , we have
Theorem 4. Equation (6) has a unique positive definite solution , and for any , the iteration converges to ; that is, .
Proof. Consider the matrix sequence (9). Let . We first suppose that . Then we get
Analogously one can prove that
Hence, the sequence is monotone increasing and is bounded from above by . Thus, the sequence has a finite positive definite limit. Moreover, the sequence is monotone decreasing and is bounded from below by . Thus, the sequence has a finite positive definite limit. Let
then for . Clearly, both and are the positive fixed points of . Then
Let ; then . Now we prove that . Assume, on the contrary, that ; then . By Lemma 3 and monotonicity of , we have
Since , then it is a contradiction to the definition of . Hence we have ; therefore . Similarly, we can get . Therefore, . Let . By Theorem 1, we know . Hence . That is, has only one positive fixed point in . In other words, the equation has only one positive definite solution. Hence . It follows that is a fixed point of . Since the positive definite solution of solves , then has only one positive definite solution. Hence is the unique fixed point of .
For any , we get By induction, we have for any Now letting on both sides of the above inequalities, we can get
Since (1) is equal to (6) when , then we know that is a positive definite solution of (1) if and only if is a positive semidefinite solution of (6). And furthermore, if and only if . Thus, we can get the following conclusions about (1).
Theorem 5. Equation (1) with has a positive definite solution for any .
Theorem 6. If is positive definite solution of (1) with , then .
Theorem 7. Let and such that . Then (1) has a unique solution in and for any , the iteration converges to ; that is, .
3. The Case of
In this section, we discuss (1) with and . In fact, must be a positive definite matrix in this case, because and because is a positive definite matrix. We consider the matrix equation which is equal to (1). Let be the set defined by We are interested in positive definite solutions of (1) in this set. A matrix is a solution of (1) if and only if it is a fixed point of the map defined by Let ; For , we have being a positive definite matrix. Then (1) turns into Let . Then (1) eventually becomes We note that (23) should have the same results as (6). Then we directly give the following conclusions without proof.
Theorem 8. Equation (1) with and has a positive definite solution for any .
Theorem 9. If is a positive definite solution of (1) with and , then .
Theorem 10. Let and such that . Then (1) has a unique solution in , and for any , the iteration converges to ; that is, .
4. Numerical Examples
We now use numerical examples to illustrate our results. All computations were performed using MATLAB, version7.01. We denote and use the stopping criterion .
Example 1. Consider (1) with , , and where the matrices and satisfy . Considering the iterative method (18) with , after 23 iterations we get the following result: and . It is not difficult to verify that .
Example 2. Consider (1) with , , and where the matrices and satisfy and . Considering the iterative method (24) with , after 7 iterations one gets an approximation to the positive definite solution . It is and . It is not difficult to verify that .
The work was supported by the National Natural Science Foundation of China (11071141), the Natural Science Foundation of Shandong Province of China (ZR2011AL018), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province, and the Project of Shandong Province Higher Educational Science and Technology Program (J11LA06 and J13LI02).
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