Finite iterative algorithm for solving a class of complex
matrix equation with two unknowns
Mokhtar A. Abdel Naby , Mohamed A. Ramadan and Ahmed M. E. Bayoumi
Department of Mathematics, Faculty of Science, Menoufia University, Shebeen El- Koom, Egypt
Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt
Abstract
This paper is concerned with an efficient iterative solution to general Sylvester-conjugate matrix equation of the form which are extension to our proposed general Sylvester-conjugate equation of the form . An iterative Algorithm is constructed to give a solution to this matrix equation. When a solution exists for this matrix equation, for any initial matrices, the solutions can be obtained within finite iterative steps in the absence of round off errors. Some lemmas and theorems are stated and proved where the iterative solutions are obtained. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm.
Keywords: General Sylvester-conjugate matrix equations; Finite iterative algorithm; Orthogonality; Inner product space; Frobenius norm.
1. Introduction
We know that matrix equation is one of the topics of very active research in computational mathematics, and a large number of papers have presented several methods for solving several matrix equations [1–5]. Ding and Chen presented the hierarchical gradient iterative algorithms for general matrix equations [6, 12] and hierarchical least squares iterative algorithms for generalized coupled Sylvester matrix equations and general coupled matrix equations [13, 14]. The hierarchical gradient iterative algorithms [6, 12] and hierarchical least squares iterative algorithms [6,14,15] for solving general (coupled) matrix equations are innovational and computationally efficient numerical ones and were proposed based on the hierarchical identification principle [13,16] which regards the unknown matrix as the system parameter
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Corresponding Author: Mohamed A. Ramadan: mramadan @ eun.eg; ramadanmohamed13@yahoo.com
matrix to be identified. In [7], the necessary and sufficient conditions for the solvability of the matrix equation , over reflexive and anti-reflexive matrices are given, and the general expression of the reflexive and anti-reflexive solutions for a solvable case is obtained. Ramadan et al. [8] introduced a complete, general and explicit solution to the Yakubovich matrix equation , with in an arbitrary form. Also with the help of the concept of Kronecker map, an explicit solution for the matrix equation was established in [9]. Zhou et al. [10] proposed gradient based iterative algorithms for solving the general coupled Sylvester matrix equations with weighted least squares solutions. In [11], a general parametric solution to a family of generalized Sylvester matrix equations arising in linear system theory is presented by using the so-called generalized Sylvester mapping which has some elegant properties.
In [17], a finite iterative algorithm for solving the generalized reflexive solution of the linear systems of matrix equations was given. In [18], solutions to the so-called coupled Sylvester-conjugate matrix equations, which include the generalized Sylvester matrix equation and coupled Lyapunov matrix equation as special cases are given. In [19], an iterative algorithm is presented for solving the extended Sylvester-conjugate matrix equation. Ramadan et. al. proposed a finite iterative solution to general Sylvester-conjugate matrix equation of the form in [20].
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 propose iterative method to obtain numerical solution to the matrix equations using iterative method. In section 4, a numerical example is given to explore the simplicity and the neatness of the presented methods.
2. Preliminaries
The following notations, definitions, lemmas and theorems will be used to develop the proposed work. We use and to denote the transpose, conjugate, conjugate transpose and the trace of a matrix respectively. We denote the set of all complex
matrices by ℂ , denote the real part of number .
Definition 1. Inner product [38]
A real inner product space is a vector space over the real field ℝ together with an inner product. i.e. With a map
ℝ
Satisfying the following three axioms for all vectors and all scalars ℝ
(1) Symmetry: .
(2) Linearity in the first argument:
,
,
(3) Positive definiteness: for all .
Two vectors are said to be orthogonal if .
The following theorem defines a real inner product on space ℂ over the field ℝ
Theorem 1. [29]
In the space ℂ over the field ℝ, an inner product can be defined as
. (2)
Proof.
(1) For ℂ , according to the properties of trace of a matrix one has
(2) For a real number a, and ℂ , one has
(3) It is well-known that for all . Thus, for all .
According to definition 1, all the above argument reveals that the space ℂ over field ℝ with the inner product defined by (2) is an inner product space.
Definition2. Frobenius Norm
The matrix norm of induced by the inner product is Frobenius norm and denoted by
.
3. Main results
In this section, we propose an iterative solution to the complex matrix equation
, (1)
where ℂ , ℂ and ℂ are given matrices, while ℂ are matrices to be determined.
The solution the matrix equation (1) is based on the following algorithm.
Algorithm I (Finite Iterative Algorithm for (1))
1. Input
2. Chosen arbitrary matrices ℂ ;
3. Set
;
;
;
4. If then stop; and are the solution; else let go to STEP 5.
5. compute
;
;
;
;
6. If ;then stop ;else let ; go to STEP 5.
To prove the convergence property of Algorithm I, we first establish the following basic properties
Lemma 1
Suppose the matrix equation (1) is consistent and let be its solution. Then, for any initial matrices , we have
(3)
Or, equivalently
Where the sequences and are generated by Algorithm I for
Proof
We apply mathematical induction to prove the conclusion
For , from Algorithm I we have
In view that are solutions of the matrix equation (1), it is easy that one can obtain from above relation
This implies that (3) holds for .
Assume that (3) holds for . That is,
Then we have to prove that the conclusion holds for . It follows from Algorithm I that
(4)
In view that is a solution of matrix equation (1), with this relation and (4) one has
Hence relation (3) holds by principle of induction.
Lemma 2
Suppose that the matrix equation (1) is consistent and the sequences , and are generated by Algorithm I with any initial matrices , such that for all then,
(5)
and for . (6)
Proof.
We apply mathematical induction
Step 1: We prove that
(7)
and for . (8)
First from Algorithm I we have
(9)
For , it follows from (9) that
.
From this last relation one has
This implies that (7) is satisfied for .
From Algorithm I we have
From this last relation one has
This implies that (8) is satisfied for .
Assume (7) and (8) hold for . From (9) and applying mathematical induction assumption, from Algorithm I we have
Thus, (7) holds for .
Also, from Algorithm I we have
This implies that (7) and (8) hold for .
Hence, relation (7) and (8) hold for all
Step2: we want to show that
(10) and (11) hold for integer . We will prove this conclusion by induction. The case of has been proven in Step 1. Now we assume that (7) and (8) holds for . The aim is to show
(12)
and (13)
First we prove the following
(14)
and (15)
By using Algorithm I, from (9) and induction assumption we have
And
Then both (14) and (15) are holds
From Algorithm I and (9), induction assumption one has
(16)
In addition, from (9) it can be shown that
(17)
Repeating (16) and (17), one can easily obtain for certain and
and
Combining these two relations with (12) and (13) implies that (8) and (9) hold for . From step (1) and (2) the conclusion holds by the principle of induction.
Remark
Lemma 1 implies that if there exist a positive number such that and but , then the matrix equation (1) is inconsistent.
With the above two lemmas, we have the following theorem.
Theorem 2[19]
If the matrix equation (1) is consistent, then a solution can be obtained within finite iteration steps by using Algorithm I for any initial matrices .
4 Numerical example
In this section, we present a numerical example to illustrate the application of our proposed methods.
Example
In this example we illustrate our theoretical result of algorithm I for solving the matrix Equation
As a special case
Given
,
, ,
, ,
, ,
Taking
. We apply Algorithm I to compute .
After iterating 14 steps we obtain
which satisfy the matrix equation
with the corresponding residual
The obtained results are presented in figure 1, where
(Residual)
From Fig. 1, it is clear that the error is becoming smaller and approaches zero as iteration number increases. This indicates that the proposed algorithm is effective and convergent.
Fig. 1. The residual and the relative error versus (iteration number)
7 Conclusions
Iterative solution for the general Sylvester-conjugate matrix equation is presented. We have proven that the iterative algorithm always converge to the solution for any initial matrices. We stated and proved some lemmas and theorems where the solutions are obtained. The obtained results show that the methods are very neat and efficient. The proposed methods are illustrated by numerical example. Example we tested using MATLAB to verify our theoretical results.
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