Positive Definite Solutions for the System of Nonlinear Matrix Equations X + A*Y-nA = I, Y + B*X-mB = I

Abstract

In this paper, some properties of the positive definite solutions for the nonlinear system of matrix equations X + A*Y-nA = I, Y + B*X-mB = I are derived. As a matter of fact, an effective iterative method to obtain the positive definite solutions of the system is established. These solutions are based on the convergence of monotone sequences of positive definite matrices. Moreover, the necessary and sufficient conditions for the existence of the positive definite solutions are obtained. Finally, some numerical results are given.

Share and Cite:

El-Sayed, S. and Al-Dubiban, A. (2014) Positive Definite Solutions for the System of Nonlinear Matrix Equations X + A*Y-nA = I, Y + B*X-mB = I. Applied Mathematics, 5, 1977-1987. doi: 10.4236/am.2014.513193.

1. Introduction

It is well known that algebraic discrete-type Riccati equations play a central role in modern control theory and signal processing. These equations arise in many important applications such as in optimal control theory, dynamic programming, stochastic filtering, statistics and other fields of pure and applied mathematics [1] -[3] .

In the last years, the nonlinear matrix equation of the form

(1.1)

where and maps from positive definite matrices into positive definite matrices is studied in many papers [4] -[8] . It is well known that Equation (1.1) with and is a special case of algebraic discrete-type Riccati equation of the form [2] [3]

(1.2)

In addition, the system (Sys.) of algebraic discrete-type Riccati equations appears in many applications [9] -[12] . Czornik and Swierniak [10] have studied the lower bounds for eigenvalues and matrix lower bound of a solution for the special case of the System:

(1.3)

where.

In the same manner, we can deduce a system of nonlinear matrix equations as matrix Equation (1.1) with and. For that, Al-Dubiban [13] have studied the system

(1.4)

which is a special case of Sys.(1.3). The author obtained sufficient conditions for existence of a positive definite solution of Sys.(1.4) and considered an iterative method to calculate the solution. Recently, similar kinds of Sys.(1.4) have been studied in some papers [14] [15] .

In this paper we consider the system of nonlinear matrix equations that can be expressed in the form:

(1.5)

where are two positive integers, X, Y are unknown matrices, I is the identity matrix, and A, B are nonsingular matrices. All matrices are defined over the complex field. The paper is organized as follows: in Section 2, we derive the necessary and sufficient conditions for the existence the solution to the Sys.(1.5). In Section 3, we introduce an iterative method to obtain the positive definite solutions of Sys.(1.5). We discuss the convergence of this iterative method. Section 4 discussed the error and the residual error. Some numerical examples are given to illustrate the efficiency for suggested method in Section 5.

The following notations are used throughout the rest of the paper. The notation means that is positive semidefinite (positive definite), denotes the complex conjugate transpose of, and is the identity matrix. Moreover, is used as a different notation for . We denote by the spectral radius of; means the eigenvalues of and respectively. The norm used in this paper is the spectral norm of the matrix, i.e. unless otherwise noted.

2. Existence Conditions of the Solutions

In this section, we will discuss some properties of the solutions for Sys.(1.5) and obtain the necessary and sufficient conditions for the existence of the solutions of the Sys.(1.5).

Theorem 1 If are the smallest and the largest eigenvalues of a solution of Sys.(1.5), respectively, and are the smallest and the largest eigenvalues of a solution of Sys.(1.5), respectively, are eigenvalues of A, B then

(1.6)

(1.7)

Proof: Let be an eigenvector corresponding to an eigenvalue of the matrix A and, be an eigenvector corresponding to an eigenvalue of the matrix B and. Since the solution of Sys.(1.5) is a positive definite solution then and.

From the Sys.(1.5), we have

i.e

Hence

Also, from the Sys.(1.5), we have

i.e

Hence

Theorem 2 If Sys.(1.5) has a positive definite solution, then

(1.8)

(1.9)

Proof: Since be a positive definite solution of Sys.(1.5), then

From the inequality, we have, therefore

From the inequality, we have, then

And from the inequality , we have, therefore

From the inequality, we have, hence

which complete the proof.

Corollary 1 If Sys.(1.5) has a positive definite solution, then

(1.10)

(1.11)

Theorem 3 Sys. (1.5) has a positive definite solution if and only if the matrices A, B have the factorization

(1.12)

where are nonsingular matrices satisfying the following system

(1.13)

In this case the solution is.

Proof: Let Sys.(1.5) has a positive definite solution, then, where are nonsingular matrices. Furthermore Sys.(1.5) can be rewritten as

Let, , then, , and Sys. (1.5) turns into Sys.

(1.13).

Conversely, if have the factorization (1.12) and satisfying Sys.(1.13), let, then are positive definite matrices , and we have

Hence Sys.(1.5) has a positive definite solution.

3. Iterative Method for the System

In this section, we will investigate the iterative solution of the Sys.(1.5). From this section to the end of the paper we will consider the matrices A, B are normal satisfing and.

Let us consider the iterative processes

(1.14)

Lemma 1 For the Sys.(1.5), we have

(1.15)

where are matrices generated from the sequences (1.14).

Proof: Since, then

Using the conditions, we obtain

Also, we have

Using the conditions, we obtain

By the same manner, we get

Further, assume that for each, we have

(1.16)

Now, by induction, we will prove

Since the two matrices A, B are normal, then by using the equalities (0.16), we have

Similarly

By using the conditions and the equalities (1.16), we have

Also,we can prove

Therefore, the equalities (1.15) are true for all.

Lemma 2 For the Sys.(1.5), we have

(1.17)

where are matrices generated from the sequences (1.14).

Proof: Since then

By using the equalities (1.15), we have

Similarly we get

Further, assume that for each it is satisfied

(1.18)

Now, by induction, we will prove

From the equalities (1.18), we have

(1.19)

By using the equalities (1.15) and (1.19), we have

By the same manner, we can prove

Therefore, the equalities (1.17) are true for all.

Theorem 4 If A, B are satisfying the following conditions:

(i)

(ii)

where, then the Sys.(1.5) has a positive definite solution.

Proof: We consider the sequences (1.14). For we have.

For we obtain

Applying the condition we obtain

i.e.

Also, we can prove that

So, assume that

(1.20)

Now, we will prove and

By using the in equalities (1.20) we have

Similarly

Also, by using the conditions and the equalities (1.20), we have

Similarly, we have

Therefore, the inequalities (1.20) are true for all. Hence is monotonically decreasing and bounded from below by the matrix. Consequently the sequence converges to a positive definite solution X. Also, the sequence is monotonically decreasing and bounded from below by the matrix

and converges to a positive definite solution. So is a positive definite solution of Sys.(1.5).

4. Estimation of the Errors

Theorem 5 If A, B are satisfying the following conditions

(i)

(ii)

then

(1.21)

(1.22)

where, are matrices generated from the sequences (1.14).

Proof: From Theorem 4 it follows that the sequences (1.14) are convergent to a positive definite solution of Sys.(1.5). We consider the spectral norm of the matrices.

According to Theorem 4 we have

Consequently

Then we get

(1.23)

Also, we have

According to Theorem 4 we have

Consequently

then we get

(1.24)

By using (1.24) in (1.23), we have

Similarly, by using (1.23) in (1.24), we have

Theorem 6 If A, B are satisfying the following conditions:

(i)

(ii)

where, and after s iterative steps of the iterative process (1.14), we have , then

(1.25)

(1.26)

Proof: Since

Taking the norm of both sides, we have

Also,

Taking the norm of both sides, we get

5. Numerical Examples

In this section the numerical examples are given to display the flexibility of the method. The solutions are computed for some different matrices A, B with different orders. In the following examples we denote X, Y the solutions which are obtained by iterative method (1.14) and

.

Example 1 Consider Sys.(1.5) with and normal matrices

and

By computation, we get

The results are given in the Table1

Example 2 Consider Sys.(1.5) with and matrices

and

By computation, we get

Table 1. Error analysis for Example 1.

Table 2. Error analysis for Example 2.

The results are given in Table2

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] Anderson Jr., W.N., Morley, T.D. and Trapp, G.E. (1990) Positive Solutions to X = A _ BX-1B*. Linear Algebra and Its Applications, 134, 53-62. http://dx.doi.org/10.1016/0024-3795(90)90005-W [2] Lancaster, P. and Rodman, L. (1995) Algebraic Riccati Equations. Oxford Science, Oxford. [3] Meini, B. (2000) Matrix Equations and Structures: Efficient Solution of Special Discrete Algebraic Riccati Equations. Proceedings of the WLSSCOO, Bulgaria, 2000. [4] Engwerda, J.C. (1993) On the Existence of a Positive Definite Solution of the Matrix Equation X + A*X-1A = I. Linear Algebra and Its Applications, 194, 91-108. http://dx.doi.org/10.1016/0024-3795(93)90115-5 [5] Hasanov, V.I. and Ivanov, I.G. (2004) Solutions and Perturbation Estimates for the Matrix Equations X ±A*X-nA = Q. Applied Mathematics and Computation, 156, 513-525. http://dx.doi.org/10.1016/j.amc.2003.08.007 [6] Hasanov, V.I. and El-Sayed, S.M. (2006) On the Positive Definite Solutions of Nonlinear Matrix Equation X + A*X-δA = Q. Linear Algebra and Its Applications, 412, 154-160. http://dx.doi.org/10.1016/j.laa.2005.06.026 [7] Ivanov, I.G. and El-Sayed, S.M. (1998) Properties of Positive Definite Solutions of the Equation X + A*X-2A = I. Linear Algebra and Its Applications, 279, 303-316. http://dx.doi.org/10.1016/S0024-3795(98)00023-8 [8] Ivanov, I.G. (2006) On Positive Definite Solutions of the Family of Matrix Equations X + A*X-nA = Q. Journal of Computational and Applied Mathematics, 193, 277-301. http://dx.doi.org/10.1016/j.cam.2005.06.007 [9] Costa, O.L.V. and Marques, R.P. (1999) Maximal and Stabilizing Hermitian Solutions for Discrete-Time Coupled Algebraic Riccati Equations. Mathematics of Control, Signals and Systems, 12, 167-195.http://dx.doi.org/10.1007/PL00009849 [10] Czornik, A. and Swierniak, A. (2001) Lower Bounds on the Solution of Coupled Algebraic Riccati Equation. Automatica, 37, 619-624. http://dx.doi.org/10.1016/S0005-1098(00)00196-5 [11] Czornik, A. and Swierniak, A. (2001) Upper Bounds on the Solution of Coupled Algebraic Riccati Equation. Journal of Inequalities and Applications, 6, 373-385. [12] Mukaidani, H., Yamamoto, S. and Yamamoto, T. (2008) A Numerical Algorithm for Finding Solution of Cross-Coupled Algebraic Riccati Equations. IEICE Transactions, 91, 682-685.http://dx.doi.org/10.1093/ietfec/e91-a.2.682 [13] Al-Dubiban, A.M. (2008) Iterative Algorithms for Computing the Positive Definite Solutions for Nonlinear Matrix Equations. Ph.D. Thesis, Riyadh University for Girls, Riyadh. [14] Al-Dubiban, A.M. (2012) Iterative Algorithm for Solving a System of Nonlinear Matrix Equations. Journal of Applied Mathematics, 2012, Article ID: 461407. [15] Al-Dubiban, A.M. (2013) On the Iterative Method for the System of Nonlinear Matrix Equations. Abstract and Applied Analysis, 2013, Article ID: 685753.