The -Reflexive Solution to System of Matrix Equations ,

Abstract

Let and be Hermitian and -potent matrices; that is, and where stands for the conjugate transpose of a matrix. A matrix is called -reflexive (antireflexive) if . In this paper, the system of matrix equations and subject to -reflexive and antireflexive constraints is studied by converting into two simpler cases: and We give the solvability conditions and the general solution to this system; in addition, the least squares solution is derived; finally, the associated optimal approximation problem for a given matrix is considered.

1. Introduction

Throughout this paper, , , and stand for the sets of all real matrices, the sets of all complex matrices, and unitary matrices, respectively. For , , , , and represent the conjugate transpose, the rank, the trace, and the Frobenius norm of a matrix , respectively. For , denotes the inner product of and . Therefore, is a complete inner product space and the norm of a matrix generated by the inner product is the Frobenius norm; that is, And denotes the Moore-Penrose inverse, namely, the unique matrix , that satisfies the following four Penrose conditions:

Let be identity matrix of size , and let be cross-identity matrix of size having the elements along the southwest-northeast diagonal and the remaining elements being zeros. The symbol is the matrix of all zeros entries (if no confusion occurs, we will omit the subscript).

A matrix is called Hermitian and -potent matrix if

Definition 1. Let and be two Hermitian and -potent matrices; we say that matrix is -reflexive (antireflexive) if

Obviously, the centrosymmetric and centroskew matrices (when ), reflexive and antireflexive matrices (when is Hermitian involution), are the natural extensions of the -reflexive and the antireflexive matrices, respectively. These kinds of matrices play important roles in information theory, linear system theory, linear estimation theory, numerical analysis theory, and so forth (see [1–3]). If and are involution, then the -reflexive and antireflexive matrices are called -symmetric and -skew symmetric matrices [4]. Moreover, the -reflexive and antireflexive matrices mentioned in [5] are also the particular case when in Definition 1. More generally, [6] has obtained the -reflexive and antireflexive solutions to the matrix equation

Investigating the classical system of matrix equations has attracted many authors’ attention. For instance, [7] gave the necessary and sufficient conditions for the consistency of (2) and [8, 9] 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 system (2) with various constraints, such as bisymmetric, Hermitian, positive semidefinite, reflexive, and generalized reflexive solutions (see [10–27]). To our knowledge, so far, there has been little investigation of the -reflexive and antireflexive solutions to (2).

Motivated by the work mentioned above, we investigate the -reflexive and antireflexive solutions to (2). We also consider the optimal approximation problem, which can be described as follows: let be a given matrix in and let be the set of all -reflexive (antireflexive) solutions to (2); find such that In many cases, the system of matrix equations (2) has no -reflexive (antireflexive) solution; hence, we need to further study its least squares solution, which can be described as follows: let () denote the set of all -reflexive (antireflexive) matrices in ; find such that

In Section 2, we present necessary and sufficient conditions for the existence of the -reflexive (antireflexive) solution to (2) and give an expression of this solution when the solvability conditions are met. In Section 3, we derive an optimal approximation solution to (3). In Section 4, we provide the least squares -reflexive (antireflexive) solution to (2). In Section 5, we give an algorithm and a numerical example to illustrate our results.

2. The -Reflexive (Antireflexive) Solution to (2)

The following lemma derived from [5, 6] characterizes the Hermitian and -potent matrices.

Lemma 2. Let and be Hermitian; then, and are -potent matrices if and only if and are idempotent (i.e., , ) when is odd, or tripotent (i.e., , ) when is even. Moreover, there exist and such that where and

In the sequel, we always think that the Hermitian and -potent matrices and are fixed and have taken the form of (5) or (6). From Lemma 2, we can see that the general -potent matrices and can be reduced to only two simpler cases: (i.e., ) and , (i.e., ). Consequently, we will discuss our problems through -reflexive (antireflexive) and -reflexive (antireflexive) constraints.

Lemma 3. Let be -reflexive if and only if or -reflexive if and only if where , , and .

Assume that and are idempotent (i.e., ) and ; then, by premultiplying and postmultiplying it by and , respectively, we get , which implies that , and, thus, In this case, the system of matrix equations (2) is inconsistent when Consequently, for the antireflexive case, we only consider the -antireflexive solution.

From (6), is -antireflexive if and only if it can be written as In addition, the following lemma is well known (see, e.g., [17]).

Lemma 4. The system of matrix equations (2) is consistent for unknown matrix if and only if In this case, the general solution is where is arbitrary.

2.1. The -Reflexive Solution to (2)

Let and be Hermitian and -potent matrices as given in (5), and partition where , , , and . Then, we have the following result.

Theorem 5. Given , , , and , let , , , be defined in (12); then, the system of matrix equations (2) is consistent for -reflexive if and only if In this case, the general solution is where is arbitrary.

Proof. From formula (7) in Lemma 3, we deduce that (2) is consistent for -reflexive can be equivalently converted into solving the following system of matrix equations: It follows from Lemma 4 that (15) is consistent if and only if all equalities in (13) hold, and the expression of the general solution is (14).

2.2. The -Reflexive Solution to (2)

Let and be Hermitian and -potent matrices as given in (6), and partition where , , , , , , , and .

Then, we have the following result.

Theorem 6. Given , , , and , let , , , , ,  , , be defined in (16); then, the system of matrix equations (2) is consistent for -reflexive if and only if In this case, the general solution is where , , and are arbitrary.

Proof. From formula (8) in Lemma 3, we deduce that (2) is consistent for -reflexive can be equivalently converted into solving the following matrix equations: It follows from Lemma 4 that (19) is consistent if and only if all equalities in (17) hold, and the expression of the general solution is (18).

Remark 7. If and are tripotent and , then and is tripotent. Hence, the -antireflexive constraint can be reduced to the -reflexive case. Similar to Theorem 6, the conclusion is omitted.

3. Solution to the Optimal Approximation Problem

This section solves the optimal approximation problem; that is, suppose that the solvability conditions of the system of matrix equations (2) in Theorem 5 and Theorem 6 hold; then, we derive the following results.

Lemma 8 (see [28]). Suppose that , and , where and . Then, if and only if , in which case

By Lemma 8, let , , , and then the Procrustes problem has a solution if and only if that is, the solution can be expressed as where , are arbitrary matrices.

Theorem 9. Let be the set of all -reflexive solutions to the system of matrix equations (2) as in Theorem 5 and let be a given matrix in . Partition Then, has an only solution which can be expressed as

Proof. It is easy to verify that the solution set is a closed and convex set in the matrix space under Frobenius norm, so the solution to the approximation problem is unique. Note that, from (14), (25), and the unitary invariance of the Frobenius norm, we get By Lemma 8, we have if and only if with , being arbitrary. Substituting into (14) deduces that (27) holds.

Theorem 10. Let be the set of all -reflexive solutions to the system of matrix equations (2) as in Theorem 6 and let be a given matrix in . Partition Then, has an only solution which can be expressed as where and

Proof. Similar to Theorem 9, the proof is omitted.

4. The Least Squares -Reflexive (Antireflexive) Solution to (2)

Lemma 11 (see [11]). Given , , , and (), (). Then, there exists a unique matrix such that and can be expressed as where

Theorem 12. Let , , , and , , , , where , , , . Assume that the singular value decomposition of , is as follows: where , , , and are unitary matrices, , , , , , , (), , , . Then, can be expressed as where and is an arbitrary matrix.

Proof. It yields from (36) that Assume that Then, we have Hence, is solvable if and only if there exist , , such that It follows from (42) that where Substituting (43) into (39), we can get that the form of elements in is (37).

Remark 13. The -reflexive and antireflexive constraint least squares problem can be reduced similar to Theorem 9; the conclusion is omitted here.

5. An Algorithm and Numerical Example

The algorithm below constructs the -reflexive solution, the optimal approximation -reflexive solution, and the least squares -reflexive solution for the problem stated in Sections 2, 3, and 4, respectively.

Algorithm 14. (1) Input , , , .
(2) Compute , , , by (12).
(3) If (13) holds, compute the -reflexive solution of (2) by (14); input and compute by (25); then, compute the solution of problem (3) according to (27).
(4) If (13) does not hold, compute the singular value decomposition of , by (36); then, compute the least squares -reflexive solution of (2) by (37).

Similar algorithms for the -antireflexive and -reflexive cases can be developed. Next, we illustrate the obtained results with two examples.

Let , and we obtain

Example 1. Suppose , , , and We can verify that (13) holds. Hence, system (2) has the -reflexive solution, which can be expressed as Given Applying Algorithm 14, we obtain the following:

Example 2. Let be the same as Example 1, and let in Example 1 be changed into We can verify that (13) does not hold. By Algorithm 14, the least squares -reflexive solution of (2) can be expressed as

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the grants from the National Natural Science Foundation of China (11571220), the Education Department Foundation of Hebei Province (QN2015218), and the Natural Science Foundation of Hebei Province (A2015403050).