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, ![](https://www.scirp.org/html/htmlimages\13-7402015x\167d4fe3-5948-4c39-8034-ca5f894b53d7.png)
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
![](https://www.scirp.org/html/htmlimages\13-7402015x\d08089c1-7a1f-49ee-98c8-3539ee7b500c.png)
![](https://www.scirp.org/html/htmlimages\13-7402015x\247a5370-8666-4a3e-909c-abf533fdb2b7.png)
![](https://www.scirp.org/html/htmlimages\13-7402015x\9c6cc3c8-35e3-4eb8-931e-b39e2a6894b7.png)
i.e
![](https://www.scirp.org/html/htmlimages\13-7402015x\c7b477a7-e48b-4ed8-a9f7-13b0261b3d67.png)
Hence
![](https://www.scirp.org/html/htmlimages\13-7402015x\a1401d34-b747-48a2-915d-144eb2a423a2.png)
Also, from the Sys.(1.5), we have
![](https://www.scirp.org/html/htmlimages\13-7402015x\d8ef94e5-c9af-434e-8631-c592f52428ff.png)
![](https://www.scirp.org/html/htmlimages\13-7402015x\5cc4b881-2e78-44ee-8f28-c190698f0d7c.png)
![](https://www.scirp.org/html/htmlimages\13-7402015x\043af17a-f0b2-4fe4-b73b-cd3621a21573.png)
i.e
![](https://www.scirp.org/html/htmlimages\13-7402015x\482c4888-03da-42a8-9603-06ba413ed521.png)
Hence
![](https://www.scirp.org/html/htmlimages\13-7402015x\303077d0-c4a9-4fd5-892f-8961cbc647a6.png)
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
![](https://www.scirp.org/html/htmlimages\13-7402015x\0bbeec7e-8907-49ce-93c0-a3097689a770.png)
From the inequality
, we have
, therefore
![](https://www.scirp.org/html/htmlimages\13-7402015x\f99e8c92-f2f6-4b8d-961b-8ad76cbc1bbf.png)
From the inequality
, we have
, then
![](https://www.scirp.org/html/htmlimages\13-7402015x\75d31c9e-17cf-45c9-ac52-58ca4db7b8af.png)
And from the inequality
, we have
, therefore
![](https://www.scirp.org/html/htmlimages\13-7402015x\cdbebd0b-efef-4a80-bbe2-0040f10c0d02.png)
From the inequality
, we have
, hence
![](https://www.scirp.org/html/htmlimages\13-7402015x\da332ddd-9529-4b6c-b9ff-989232fcd06a.png)
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
![](https://www.scirp.org/html/htmlimages\13-7402015x\2e06fe75-7907-4350-8a43-42cf1ed5525b.png)
![](https://www.scirp.org/html/htmlimages\13-7402015x\31e2be14-9904-48bc-af43-1c010e4f2b4b.png)
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
![](https://www.scirp.org/html/htmlimages\13-7402015x\75e6afae-18c3-41ba-8565-c9da5e7e85e7.png)
![](https://www.scirp.org/html/htmlimages\13-7402015x\31c296bb-faf0-4fe8-aa98-7aa687fecef6.png)
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
![](https://www.scirp.org/html/htmlimages\13-7402015x\c683c381-0ad3-4f41-9b02-333971b115e5.png)
Using the conditions
, we obtain
![](https://www.scirp.org/html/htmlimages\13-7402015x\a84ca811-1af5-4c1a-aefb-754e42ea6fdc.png)
Also, we have
![](https://www.scirp.org/html/htmlimages\13-7402015x\b1870371-dc2f-432a-ad61-c46f4cf44028.png)
Using the conditions
, we obtain
![](https://www.scirp.org/html/htmlimages\13-7402015x\b05e6b37-c0d6-41c9-8a0b-b01ac702f1c4.png)
By the same manner, we get
![](https://www.scirp.org/html/htmlimages\13-7402015x\ac03587a-9df7-4f8b-86c5-f80fee0c83a3.png)
Further, assume that for each
, we have
(1.16)
Now, by induction, we will prove
![](https://www.scirp.org/html/htmlimages\13-7402015x\454efa28-2873-4e81-b92c-98fc82573676.png)
Since the two matrices A, B are normal, then by using the equalities (0.16), we have
![](https://www.scirp.org/html/htmlimages\13-7402015x\99291743-245f-4bb4-ac8e-6ce900f6d99d.png)
Similarly
![](https://www.scirp.org/html/htmlimages\13-7402015x\44e4d944-5cdc-452c-b813-073641e81f09.png)
By using the conditions
and the equalities (1.16), we have
![](https://www.scirp.org/html/htmlimages\13-7402015x\c339f5df-666d-4edd-a5f4-5766ee8cef1b.png)
Also,we can prove
![](https://www.scirp.org/html/htmlimages\13-7402015x\89d76ab0-fce2-4df7-8cbf-4e1d14e3602a.png)
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 ![](https://www.scirp.org/html/htmlimages\13-7402015x\3c222783-ff39-4b0d-af0a-c3238230c143.png)
By using the equalities (1.15), we have
![](https://www.scirp.org/html/htmlimages\13-7402015x\52ddfda7-dc8a-467e-a578-2a5178c06d45.png)
Similarly we get
![](https://www.scirp.org/html/htmlimages\13-7402015x\e41d7238-0f04-49e3-822a-6e9e9d77a257.png)
Further, assume that for each
it is satisfied
(1.18)
Now, by induction, we will prove
![](https://www.scirp.org/html/htmlimages\13-7402015x\53b3d69f-9993-4b05-a456-8653c4292ca9.png)
From the equalities (1.18), we have
(1.19)
By using the equalities (1.15) and (1.19), we have
![](https://www.scirp.org/html/htmlimages\13-7402015x\910245a4-6bd6-4034-9403-3d8448f631c5.png)
By the same manner, we can prove
![](https://www.scirp.org/html/htmlimages\13-7402015x\7072de8a-7dea-4034-af8d-f702f8202b81.png)
Therefore, the equalities (1.17) are true for all
.
Theorem 4 If A, B are satisfying the following conditions:
(i) ![](https://www.scirp.org/html/htmlimages\13-7402015x\e83b755e-69b9-4168-97eb-bbff2ddfe5cb.png)
(ii) ![](https://www.scirp.org/html/htmlimages\13-7402015x\8467088a-ab6a-4eb7-bac9-2df7f0c829af.png)
where
, then the Sys.(1.5) has a positive definite solution.
Proof: We consider the sequences (1.14). For
we have
.
For
we obtain
![](https://www.scirp.org/html/htmlimages\13-7402015x\3189d60c-7e00-4fb0-a167-517f0e29f967.png)
Applying the condition
we obtain
![](https://www.scirp.org/html/htmlimages\13-7402015x\4b3d52f9-9d00-440a-82fe-8803ce1c5191.png)
i.e.
![](https://www.scirp.org/html/htmlimages\13-7402015x\d09d05ff-90d4-4cb9-8dac-179f1a50e15a.png)
Also, we can prove that
![](https://www.scirp.org/html/htmlimages\13-7402015x\b63b3c0a-23ea-4235-ae9c-c77eb84a9357.png)
So, assume that
(1.20)
Now, we will prove
and ![](https://www.scirp.org/html/htmlimages\13-7402015x\3b8c7aa3-f000-4032-bd6e-f3d6ce641770.png)
By using the in equalities (1.20) we have
![](https://www.scirp.org/html/htmlimages\13-7402015x\39b5fbc3-1a7b-440a-9773-69cc7ca823a8.png)
Similarly
![](https://www.scirp.org/html/htmlimages\13-7402015x\b0647810-1c86-4a13-935a-b931a5b41c49.png)
Also, by using the conditions
and the equalities (1.20), we have
![](https://www.scirp.org/html/htmlimages\13-7402015x\f1ca30c8-60da-48ce-a01c-0fca9132033f.png)
Similarly, we have
![](https://www.scirp.org/html/htmlimages\13-7402015x\1fa4687c-c56c-4136-8827-6689027bc9c7.png)
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 ![](https://www.scirp.org/html/htmlimages\13-7402015x\e1d31687-e911-458b-902c-a86d04a00904.png)
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) ![](https://www.scirp.org/html/htmlimages\13-7402015x\772cc815-3dda-4340-a95b-9bb703690961.png)
(ii) ![](https://www.scirp.org/html/htmlimages\13-7402015x\1d432598-fb99-4813-8a5b-23d3751530a9.png)
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
.
![](https://www.scirp.org/html/htmlimages\13-7402015x\117e9d62-f3ae-489a-9d13-998d799ab75f.png)
According to Theorem 4 we have
![](https://www.scirp.org/html/htmlimages\13-7402015x\ceed066e-f1d8-43cd-808c-64f542b97702.png)
Consequently
![](https://www.scirp.org/html/htmlimages\13-7402015x\22324570-011c-4b9c-964a-e69941b94330.png)
Then we get
(1.23)
Also, we have
![](https://www.scirp.org/html/htmlimages\13-7402015x\2563cdf8-7fcf-4935-9b52-55fd48fe71c1.png)
According to Theorem 4 we have
![](https://www.scirp.org/html/htmlimages\13-7402015x\5808a32d-cbf0-489f-8b92-708993fa9fa2.png)
Consequently
![](https://www.scirp.org/html/htmlimages\13-7402015x\fca985b5-898e-4912-8956-3e66d796151d.png)
then we get
(1.24)
By using (1.24) in (1.23), we have
![](https://www.scirp.org/html/htmlimages\13-7402015x\a22a7952-e24e-4331-b59c-732d83e93dfa.png)
Similarly, by using (1.23) in (1.24), we have
![](https://www.scirp.org/html/htmlimages\13-7402015x\4220e1c8-6bfb-4832-a3fc-dd04b51a959c.png)
Theorem 6 If A, B are satisfying the following conditions:
(i) ![](https://www.scirp.org/html/htmlimages\13-7402015x\ac1d1c38-ea97-441f-932f-6b29648fed1a.png)
(ii) ![](https://www.scirp.org/html/htmlimages\13-7402015x\38176eb8-e7d6-4e60-9d75-bc08affa3dcb.png)
where
, and after s iterative steps of the iterative process (1.14), we have
, then
(1.25)
(1.26)
Proof: Since
![](https://www.scirp.org/html/htmlimages\13-7402015x\73f97262-f6d7-4d65-b42f-7c51572f1c29.png)
Taking the norm of both sides, we have
![](https://www.scirp.org/html/htmlimages\13-7402015x\85c052a2-2f3c-4f49-baf3-92d68b7ce46c.png)
Also,
![](https://www.scirp.org/html/htmlimages\13-7402015x\003a71fe-bb05-4f21-8cb2-36bb55e14952.png)
Taking the norm of both sides, we get
![](https://www.scirp.org/html/htmlimages\13-7402015x\e7a3c3c6-e03f-4556-b81c-b0ab9fdf54ce.png)
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
![](https://www.scirp.org/html/htmlimages\13-7402015x\6bb4e1ab-8f6a-4c2e-835e-1a8f774ace48.png)
and
![](https://www.scirp.org/html/htmlimages\13-7402015x\d789fe1e-fbc2-44be-ba5f-8ca0381046ec.png)
By computation, we get
![](https://www.scirp.org/html/htmlimages\13-7402015x\846a4d78-4dc7-4836-83ac-1ba166ff8d3b.png)
![](https://www.scirp.org/html/htmlimages\13-7402015x\3f985e08-fead-4b1a-ae1f-ea2b55d1efd5.png)
![](https://www.scirp.org/html/htmlimages\13-7402015x\f94f9fed-9d87-4731-bb88-477a04321af0.png)
![](https://www.scirp.org/html/htmlimages\13-7402015x\ba69f1e4-7d54-4d8c-991d-c53fae41535d.png)
The results are given in the Table1
Example 2 Consider Sys.(1.5) with
and matrices
![](https://www.scirp.org/html/htmlimages\13-7402015x\25506d4b-3d48-4d8e-981e-59e067223ff8.png)
and
![](https://www.scirp.org/html/htmlimages\13-7402015x\e8f62c1d-edd7-4fa6-adea-aae7771bd992.png)
By computation, we get
![](https://www.scirp.org/html/htmlimages\13-7402015x\83c156a7-7090-4b32-bbbf-a2d0f4b03809.png)
![](Images/Table_Tmp.jpg)
Table 1. Error analysis for Example 1.
![](Images/Table_Tmp.jpg)
Table 2. Error analysis for Example 2.
![](https://www.scirp.org/html/htmlimages\13-7402015x\b5e15e49-1bb4-4f82-9655-6dbbccbb7535.png)
![](https://www.scirp.org/html/htmlimages\13-7402015x\0f685078-f4ad-4404-8837-843af821c095.png)
![](https://www.scirp.org/html/htmlimages\13-7402015x\68ad8249-08ef-46e8-8769-35c5d0aedf52.png)
The results are given in Table2