Abstract
This paper considers a new method to solve the first-order and second-order nonhomogeneous generalized Sylvester matrix equations and , respectively, where , and are the arbitrary real known matrices and and are the matrices to be determined. An explicit solution for these equations is proposed, based on the orthogonal reduction of the matrix to an upper Hessenberg form . The technique is very simple and does not require the eigenvalues of matrix to be known. The proposed method is illustrated by numerical examples.
1. Introduction
Consider the following two homogeneous generalized Sylvester matrix equations:
Matrix equation (1) is called a first-order homogeneous generalized Sylvester matrix equation that is closely related to many problems in linear systems theory, such as eigenstructure assignment [1–5] and control of systems with input constraints [6]. Second-order homogeneous generalized Sylvester matrix equation (2) has found applications in many control problems, for example, pole assignment [7–9] and eigenstructure assignment [10, 11].
As a generalization of the above matrix equations, we have considered the following nonhomogeneous generalized Sylvester matrix equation:where , , and are the known matrices, while and need to be determined. Several authors have studied different methods for matrix equation (3) (see for example, Song and Chen, [12], Ramadan et al. [13], Duan [14], and Wu et al. [15]). The second-order nonhomogeneous Sylvester matrix equation was introduced by Duan [16]. Recently, Ramadan et al. [17] developed the Hessenberg method to solve the Sylvester matrix equation by reducing only one coefficient matrix to a block upper Hessenberg form. The main goal of this paper is to present algorithms for solving well-known nonhomogeneous generalized Sylvester matrix equations (3) and (4). The proposed algorithm differs from the preceding standard methods; in our algorithms, is only reduced to an unreduced upper Hessenberg matrix , and eigenvalues of matrix must be distinct, because if any eigenvalue of repeats, then it is defective. An unreduced Hessenberg matrix is always nonsingular, so must be nonsingular. Throughout this paper, the notation (GSME) is used for generalized Sylvester matrix equation and we assume that , , , and .
2. The Proposed Method for
Consider the following nonhomogeneous generalized first-order Sylvester matrix equation:where , and are the known matrices, while are to be determined. The following lemma plays a vital role in this paper.
Lemma 1. (see [18, 19]). Let be an unreduced upper Hessenberg matrix and let be the successive columns of a matrix that commutes with ; then,(i) can be chosen arbitrarily.(ii)The columns through can be computed recursively by the following formula:Algorithm 1 constructs an unknown matrix and compute matrix for the following matrix equation:orwhere are the columns of , is the zero vector, is the unknown vector, and is the known real matrix.
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Theorem 1. The solution of matrix equation (5) is and , where , and are the known matrices and the matrix is generated as in Lemma 1.
Proof. Matrix equation (5) can be rewritten in the following form:That is,where is an unreduced upper Hessenberg matrix, , and is an orthogonal similarity transformation. The matrix is generated by (6), and matrix equation (7) is multiplied by to getBy using Lemma 1, assume that , and , where and are computed from Algorithm 1. Then, we recover the original problem via the relations , , , and , and then, the solution of (5) is and (see Algorithm 2).
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3. The Proposed Method of
Consider the following second-order nonhomogeneous generalized Sylvester matrix equation:where , and are the known matrices, while and are to be determined.
The following algorithm constructs an unknown matrix and computes matrix for the following matrix equation:
Putting in (13), we getor
Since , then
Both , , and are columns of , , and , respectively. is a real matrix, where is the zero vector and and are the unknown vectors. From (15) and (16), we design Algorithm 3.
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Theorem 2. The solution of matrix equation (12) is and , where , and are the known matrices, and the matrix is generated as in Lemma 1.
Proof. Matrix equation (12) can be rewritten in the Hessenberg form as follows:That iswhere and are to be determined, while is an unreduced upper Hessenberg matrix, , and is an orthogonal similarity transformation. The main idea is to find a matrix and matrix for a new matrix equation (13) as shown in Algorithm 3, where is known as real matrix while the matrix is generated by (6). Multiply matrix equation (13) by ; then,By using Lemma 1 and assuming that , , and we haveWe then recover the original problem via the relations , , and ; then, the solution of (12) is and (see Algorithm 4).
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4. Numerical Examples
In this section, we present two numerical examples to illustrate the application of our proposed method.
Example 1. We solve first-order GSME (3) where and are random matrices. The accuracy of the proposed method is reported for different values of , when , and for different values of , when , as in Table 1.
Example 2. We solve the second-order GSME (4) where and are random matrices. The accuracy of the proposed method is reported for different values of , when , and for different values of , when , as in Table 2.
5. Discussion
The accuracy of the proposed methods is remarkably dependent on . If is too large, one of the entries of may have a tendency to be zero, which affects singularity of the matrix. Thus, the smaller the value of , the greater the precision. In previous studies of GSME [12–15] and [16], the working hypotheses that size of known matrices are very small. However, in our method, is very large.
6. Conclusion
In this study, the solution of GSME and , where , and are arbitrary real known matrices and and are the matrices to be determined, is investigated. With the help of orthogonal similarity transformation and reduction to Hessenberg form, some good results are obtained. The proposed techniques are tested by solving two test problems where the accuracy is seen to be highly remarkable.
7. Open Problem
Extend the Hessenberg method to solve Sylvester quaternion matrix equation [20] and coupled matrix equation [21].
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This project was supported by the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University under the research project #2017/01/7944. The authors are very grateful to Professor Reny George for the proofreading of the research.