Advances in Matrices, Finite and Infinite, with ApplicationsView this Special Issue
On the Hermitian -Conjugate Solution of a System of Matrix Equations
Let be an by nontrivial real symmetric involution matrix, that is, . An complex matrix is termed -conjugate if , where denotes the conjugate of . We give necessary and sufficient conditions for the existence of the Hermitian -conjugate solution to the system of complex matrix equations and present an expression of the Hermitian -conjugate solution to this system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least squares Hermitian -conjugate solution with the least norm to this system mentioned above is considered. The representation of such solution is also derived. Finally, an algorithm and numerical examples are given.
Throughout, we denote the complex matrix space by , the real matrix space by , and the set of all matrices in with rank by . The symbols , and stand for the identity matrix with the appropriate size, the conjugate, the transpose, the conjugate transpose, the Moore-Penrose generalized inverse, and the Frobenius norm of , respectively. We use to denote the backward matrix having the elements along the southwest diagonal and with the remaining elements being zeros.
Recall that an complex matrix is centrohermitian if . Centrohermitian matrices and related matrices, such as k-Hermitian matrices, Hermitian Toeplitz matrices, and generalized centrohermitian matrices, appear in digital signal processing and others areas (see, [1–4]). As a generalization of a centrohermitian matrix and related matrices, Trench  gave the definition of -conjugate matrix. A matrix is -conjugate if , where is a nontrivial real symmetric involution matrix, that is, and . At the same time, Trench studied the linear equation for -conjugate matrices in , where are known column vectors.
Investigating the matrix equation with the unknown matrix being symmetric, reflexive, Hermitian-generalized Hamiltonian, and repositive definite is a very active research topic [6–14]. As a generalization of (1.1), the classical system of matrix equations has attracted many author’s attention. For instance,  gave the necessary and sufficient conditions for the consistency of (1.2), [16, 17] derived an expression for the general solution by using singular value decomposition of a matrix and generalized inverses of matrices, respectively. Moreover, many results have been obtained about the system (1.2) with various constraints, such as bisymmetric, Hermitian, positive semidefinite, reflexive, and generalized reflexive solutions (see, [18–28]). To our knowledge, so far there has been little investigation of the Hermitian -conjugate solution to (1.2).
Motivated by the work mentioned above, we investigate Hermitian -conjugate solutions to (1.2). We also consider the optimal approximation problem where is a given matrix in and the set of all Hermitian -conjugate solutions to (1.2). In many cases the system (1.2) has not Hermitian -conjugate solution. Hence, we need to further study its least squares solution, which can be described as follows: Let denote the set of all Hermitian -conjugate matrices in : Find such that
In Section 2, we present necessary and sufficient conditions for the existence of the Hermitian -conjugate solution to (1.2) and give an expression of this solution when the solvability conditions are met. In Section 3, we derive an optimal approximation solution to (1.3). In Section 4, we provide the least squares Hermitian -conjugate solution to (1.5). In Section 5, we give an algorithm and a numerical example to illustrate our results.
2. -Conjugate Hermitian Solution to (1.2)
In this section, we establish the solvability conditions and the general expression for the Hermitian -conjugate solution to (1.2).
We denote and the set of all -conjugate matrices and Hermitian -conjugate matrices, respectively, that is, where is nontrivial real symmetric involution matrix.
Throughout this paper, we always assume that the nontrivial symmetric involution matrix is fixed which is given by (2.2) and (2.3). Now, we give a criterion of judging a matrix is -conjugate Hermitian matrix.
Theorem 2.1. A matrix if and only if there exists a symmetric matrix such that , where with being the same as (2.2).
Proof. If , then . By (2.2),
which is equivalent to
Substituting (2.7) into (2.6), we obtain
Hence, are real matrices. If we denote , then by (2.7)
Let , and
Then, can be expressed as , where is unitary matrix and is a real matrix. By
we obtain .
Conversely, if there exists a symmetric matrix such that , then it follows from (2.3) that that is, .
Theorem 2.1 implies that an arbitrary complex Hermitian -conjugate matrix is equivalent to a real symmetric matrix.
Lemma 2.2. For any matrix , where
Proof. For any matrix , it is obvious that , where are defined as (2.13). Now, we prove that the decomposition is unique. If there exist such that , then It follows from are real matrix that Hence, holds, where are defined as (2.13).
Lemma 2.3 (Theorem 1 in ). Let . The of matrix is as follows where and are orthogonal matrices, . Then, (1.1) has a symmetric solution if and only if In that case, it has the general solution where is an arbitrary symmetric matrix.
By Lemma 2.3, we have the following theorem.
Theorem 2.4. Given , and . Let , be defined in (2.17), (2.20), respectively. Assume that the of is as follows where and are orthogonal matrices, . Then, system (1.2) has a solution in if and only if In that case, it has the general solution where is an arbitrary symmetric matrix.
3. The Solution of Optimal Approximation Problem (1.3)
Theorem 3.1. Given , and . Assume is nonempty, then the optimal approximation problem (1.3) has a unique solution and
Proof. Since is nonempty, has the form of (2.27). By Lemma 2.2, can be written as where According to (3.2) and the unitary invariance of Frobenius norm We get Then, is consistent if and only if there exists such that For the orthogonal matrix Therefore, is equivalent to Substituting (3.9) into (2.27), we obtain (3.1).
4. The Solution of Problem (1.5)
In this section, we give the explicit expression of the solution to (1.5).
Theorem 4.1. Given , and . Let , and be defined in (2.17), (2.20), respectively. Assume that the of is as (2.25) and system (1.2) has not a solution in . Then, can be expressed as where is an arbitrary symmetric matrix.
Proof. It yields from (2.17)–(2.21) and (2.25) that Assume that Then, we have Hence, is solvable if and only if there exist such that It follows from (4.6) that Substituting (4.7) into (4.3) and then into (2.16), we can get that the form of elements in is (4.1).
Theorem 4.2. Assume that the notations and conditions are the same as Theorem 4.1. Then, if and only if
5. An Algorithm and Numerical Example
Base on the main results of this paper, we in this section propose an algorithm for finding the solution of the approximation problem (1.3) and the least squares problem with least norm (1.5). All the tests are performed by MATLAB 6.5 which has a machine precision of around .
Algorithm 5.1. Input .
Compute by (2.17) and (2.20).
Compute the singular value decomposition of with the form of (2.25).
If (2.26) holds, then input and compute the solution of problem (1.3) according (3.1), else compute the solution to problem (1.5) by (4.9).
This research was supported by the Grants from the Key Project of Scientific Research Innovation Foundation of Shanghai Municipal Education Commission (13ZZ080), the National Natural Science Foundation of China (11171205), the Natural Science Foundation of Shanghai (11ZR1412500), the Ph.D. Programs Foundation of Ministry of Education of China (20093108110001), the Discipline Project at the corresponding level of Shanghai (A. 13-0101-12-005), Shanghai Leading Academic Discipline Project (J50101), the Natural Science Foundation of Hebei province (A2012403013), and the Natural Science Foundation of Hebei province (A2012205028). The authors are grateful to the anonymous referees for their helpful comments and constructive suggestions.
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