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