Journal of Applied Mathematics

Volume 2013 (2013), Article ID 537520, 7 pages

http://dx.doi.org/10.1155/2013/537520

Research Article

Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution for the Matrix Equation

Naglaa M. El-Shazly

Department of Mathematics, Faculty of Science, Menoufia University, Shebin El-Koom, Egypt

Received 19 December 2012; Revised 26 March 2013; Accepted 14 May 2013

Academic Editor: Theodore E. Simos

Copyright © 2013 Naglaa M. El-Shazly. 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.

In this paper necessary and sufficient conditions for the matrix equation to have a positive definite solution are derived, where , is an identity matrix, are nonsingular real matrices, and is an odd positive integer. These conditions are used to propose some properties on the matrices , . Moreover, relations between the solution and the matrices are derived.

1. Introduction

Consider the nonlinear matrix equation

where is an identity matrix, are nonsingular real matrices, and is an odd positive integer ( stands for the conjugate transpose of the matrix ). Several authors [1–13] have studied the existence, the rate of convergence, as well as the necessary and sufficient conditions of the existence of positive definite solutions of similar kinds of nonlinear matrix equations.

This paper is organized as follows. First, in Section 2, we introduce some notations, lemmas, and theorems that will be needed to develop this work. In Section 3, we present necessary and sufficient conditions for the existence of a positive definite solution of (1). In Section 4, some applications of the obtained results as well as relations between the solution and the matrices are given.

2. Preliminaries

In this section, we present some notations, lemmas, and theorems that will be needed to develop this paper. We denote by , , and the spectral radius of , the eigenvalues of , and the Euclidean norm of , respectively. The notation indicates that is positive definite (semidefinite).

Lemma 1. Let and be two arbitrary compatible matrices. Then,

The proof of this lemma is given in [14, Lemma 6].

Theorem 2 (Cholesky Decomposition). If is an symmetric positive definite matrix, then there exists a triangular matrix with positive diagonal entries such that .

For proof, see [15, page 141].

Theorem 3 (C-S decomposition [15, page 77] and [16, page 37]). If is an orthogonal matrix where , then there exist orthogonal matrices , , , and diagonal matrices , such that

where .

3. Necessary and Sufficient Conditions

In this section, we derive both necessary and sufficient conditions for the existence of a positive definite solution of the nonlinear matrix equation (1).

Theorem 4. The matrix equation (1) has a solution (symmetric and positive definite) if and only if , , admit the following factorization:

where is a nonsingular square matrix and the columns of

are orthonormal. In this case, we take as a solution of the matrix equation (1).

Proof. If (1) has a solution , then one can write as for some nonsingular matrix . In particular, can be chosen to be triangular using Theorem 2; thus, we have

Then (1) can be rewritten as

or equivalently

Let

Then

and (8) means that the columns of

are orthonormal.

Conversely, suppose that , have the decomposition (4). Set . Then

That is, is a solution to the matrix equation (1).

Theorem 5. The matrix equation (1) has a solution if and only if there exist orthogonal matrices and and diagonal matrices and with such that

In this case, is a solution of (1).

Proof. Suppose that the matrix equation (1) has a solution. Then from Theorem 4,   , , have the form

Since

is column-orthonormal, it can be extended to an orthogonal matrix

and applying Theorem 3, there exist orthogonal matrices and diagonal matrices and such that

where .

This decomposition is a generalization for Theorem 3 in [15, page 77] and [16, page 37]. It is clear that if we set , we get the special case of that theorem.

Thus , and

Then (4) can be rewritten as follows:

Conversely, suppose that , have the decomposition (13). Let . Then

This shows that is a solution to the matrix equation (1).

4. Main Results

In this section, we explain some properties of (1) and we obtain relations between the solution and the matrices .

Theorem 6. If the matrix equation (1) has a positive definite solution , then .

Proof. Suppose that (1) has a solution. Then by Theorem 5, has the decomposition (13).

Then

Since each , , is orthogonal, also is orthogonal. So

Since we choose , then we have , and therefore

Thus,

The last inequality follows from the fact , which yields that and .

In the following theorem, using the necessary and sufficient conditions which we have derived, we state and prove some inequalities in case of the matrix equation (1) having positive definite solution.

Theorem 7. Suppose that the matrix equation (1) has a positive definite solution ; then the following hold:

(i),(ii),(iii),(iv),(v).

Proof. (i) By Theorem 4, we have and . Then

since , , and , and, hence, . Thus, part (i) is proved.

(ii) Replacing by , we get

This completes the proof of (ii).

(iii) Using Theorem 6, we get

Since we choose , then we have , and therefore , ; then

Let , . Then . Thus

(iv) Appling Lemma 1, we get

(v) Consider

Therefore, .

5. Conclusion

In this paper, both necessary and sufficient conditions for the nonlinear matrix equation to have a positive definite solution are given, where , are nonsingular real matrices, and is an odd positive integer. Some properties of this matrix equation are explained. Also, relations between the positive definite solution and the matrices , , are presented.

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