Abstract

For a quaternion matrix A, we denote by the matrix obtained by applying ϕ entrywise to the transposed matrix where ϕ is a nonstandard involution of quaternions. A is said to be ϕ-Hermitian or ϕ-skew-Hermitian if or , respectively. In this paper, we give a complete characterization of the nonstandard involutions ϕ of quaternions and their conjugacy properties; then we establish a new real representation of a quaternion matrix. Based on this, we derive some necessary and sufficient conditions for the existence of a ϕ-Hermitian solution or ϕ-skew-Hermitian solution to the quaternion matrix equation . Moreover, we give solutions of the quaternion equation when it is solvable.

1. Introduction

The definitions of ϕ-Hermitian and ϕ-skew-Hermitian quaternion matrices were first introduced by Rodman (Definition 3.6.1 in [1]). As a special case of ϕ-(skew)-Hermitian matrix, the η-(anti)-Hermitian matrix first arises in widely linear modelling and has important applications over independent component analysis [2] and convergence analysis in statistical signal processing [3]. Recently, it was widely investigated (e.g., [417]). For instance, Horn and Zhang [15] gave a singular value decomposition for η-Hermitian matrix. Yuan et al. [16, 17] derived least squares η-(anti)-Hermitian solution of some classic quaternion matrix equations, while there have been a few papers to consider the ϕ-Hermitian and ϕ-skew-Hermitian solutions of some quaternion matrix equations. For example, the ϕ-Hermitian solutions of the mixed pairs of Sylvester matrix equations,were given by using Moore–Penrose inverses [18]. The ϕ-skew-Hermitian solution of the quaternion matrix equations,was discussed through matrix decomposition [19]. Very recently, some practical necessary and sufficient conditions for the existence of a solution of the system of quaternion matrix equationswere given by [20] in terms of ranks and Moore–Penrose inverses. As we know that real (complex) representation is a usual method to address the problems of quaternion matrix theory, it enables us to convert problems over quaternions into the problems over real (complex) number field. Up to now, there are many existing real (complex) representations. But, as far as we know, none of those can well preserve the structure of ϕ-(skew)-Hermitian matrix. Thus, we define a new real representation of a quaternion matrix, which can map a ϕ-Hermitian matrix or ϕ-skew-Hermitian matrix into a skew-symmetric or symmetric matrix. Motivated by the above works and its important applications, we will present a new real representation of a quaternion matrix to discuss the ϕ-Hermitian or ϕ-skew-Hermitian solution of the quaternion matrix equation .

The paper is organized as follows. In Section 2, we give a complete characterization of the nonstandard involutions of and their conjugacy properties. In Section 3, we present a new real representation of quaternion matrix, which could well preserve the structure of ϕ-(skew)-Hermitian matrix. In Section 4, we discuss the existence of the ϕ-(skew)-Hermitian solution of the quaternion matrix equation and derive the solutions when it is solvable.

2. A Complete Characterization of the Nonstandard Involutions of and Their Conjugacy Properties

In this section, we give a complete characterization of the nonstandard involutions of and their conjugacy properties.

Let and stand, respectively, for the real number field and the set of all matrices over the real quaternion algebra

The symbols stand for the identity matrix, the zero matrix with appropriate size, the transpose of A, respectively. The Moore–Penrose inverse of a real matrix A is defined to be the unique matrix such that(i)(ii)(iii)(iv)

Furthermore, and stand for the projectors and induced by A, respectively. For any , is defined as the usual conjugate transpose of A.

In [1], Rodman defined the involution over as follows.

Definition 1 (involution) [1]. A map ϕ: is called an antiendomorphism if for all and for all An antiendomorphism ϕ is called an involution if for every

The matrix representations of involutions are given in the following lemma.

Lemma 1 [1]. Let ϕ be an antiendomorphism of . Assume that ϕ does not map into zero. Then, ϕ is one-to-one and onto ; thus, ϕ is in fact an antiautomorphism. Moreover, ϕ is real linear and can be represented as a real matrix with respect to the basis . Then, ϕ is an involution if and only ifwhere either or T is a real orthogonal symmetric matrix with eigenvalues of .

By Lemma 1, involutions can be classified into two classes: the standard involution and the nonstandard involution, as defined below.

Definition 2 (standard involution) [1]. An involution ϕ is standard if .

Definition 3 (nonstandard involution) [1]. An involution ϕ is nonstandard ifwhere T is a real orthogonal symmetric matrix with eigenvalues of .
In this paper, we only consider the nonstandard involution. The following theorem shows that each nonstandard involution is in the form of , where is unit and pure imaginary and stands for the conjugate transpose of the quaternion a.

Theorem 1. A map is a nonstandard involution if and only if it is of the form , where is unit and pure imaginary, that is, and . Moreover, let , then the matrix representation of such a nonstandard involution is where is a real Householder matrix.

Proof. First, for each with and , we show that defines a nonstandard involution. By direct calculation, we have , andHence, the matrix representation of ϕ with respect to is whereThen, T is a real Householder matrix, which has eigenvalues of and is symmetric. Hence, ϕ is a nonstandard involution by Lemma 1.
Second, Lemma 1 shows that every nonstandard involution ϕ has the matrix representation where T is an orthogonal matrix with eigenvalues of . Let . Then, there exists a real orthogonal matrix U such thatwhere is the first column of U and is a real unit vector. Denote and . Then, ϕ is of the form , as desired.

We give two examples to understand the nonstandard involution.

Example 1 [4]. Let and . Then, the map is a nonstandard involution, where . Upon computation, we have

Example 2 [18]. Let and Then, the map is a nonstandard involution, where Upon computation, we haveThe product of two nonstandard involutions is in general not an involution. However, every product of the form , where δ and ϕ are nonstandard involutions, is again a nonstandard involution. Moreover, Theorem 1 implies that all nonstandard involutions are conjugate via nonstandard involutions, as shown below.

Theorem 2. Every nonstandard involution ϕ of is conjugate via a nonstandard involution to where . In other words, there exists a nonstandard involution δ such that

Proof. The matrix representation of is By Theorem 1, suppose the nonstandard involution ϕ has the matrix representation , where is a real unit vector. We hope to find a real Householder matrix , where is a real unit vector, such thatIn other words, we need , explicitly,If , then , and (14a) gives ; we simply choose such that . Otherwise , we may choose to satisfy (14a) and then solve and by (14b) and (14c). The equations (14a)–(14c) imply thatIn both cases above, we will have . Now let δ be the nonstandard involution with the matrix representation Then, (12) holds.

For a given nonstandard involution ϕ over and Rodman [1] denotes by the matrix obtained by applying ϕ entrywise to the transposed matrix . Then, by Theorem 1, the general form of follows.

Lemma 2. Let ϕ be a nonstandard involution. Then, there exists a unit pure imaginary quaternion η such that is in the form ofWe make an example in the following.

Example 3. If for , thenThe following are the definitions of ϕ-(skew)-Hermitian matrix. They are the special cases of ϕ-(skew)-Hermitian.

Definition 4 (ϕ-Hermitian or ϕ-skew-Hermitian) [1]. is said to be ϕ-Hermitian or ϕ-skew-Hermitian if or , where ϕ is a nonstandard involution.
From Lemma 2 and Definition 4, there follows a equivalent description of Definition 4: let ϕ is a nonstandard involution, that is, , where is unit and pure imaginary. Then, is said to be ϕ-Hermitian or ϕ-skew-Hermitian if or . It will play important role in solving our ϕ-Hermicity problem.

Example 4. For , is said to be η-Hermitian or η-anti-Hermitian if or , where . Obviously, η-Hermitian (η-anti-Hermitian) matrix is a special case of ϕ-Hermitian (ϕ-skew-Hermitian) matrix.

3. New Real Representation of Quaternion Matrix

Real representation method is a powerful tool to study the quaternions. There are several existing real (complex) representations of quaternion matrix, for example, the following real representation in [21, 22].

Definition 5. For defineThe above real representation of a quaternion matrix can map an Hermitian matrix or a skew-Hermitian matrix over into a symmetric matrix or skew-symmetric matrix over . In this section, we aim at exploring the ϕ-(skew)-Hermitian solutions of the quaternion matrix . To well preserve the structure of ϕ-(skew)-Hermitian matrix, in this section, we present a new representation of quaternion matrix basing on the above one. Let be a unit and pure imaginary quaternion. Now, applying the product rule to the matrix equalitywe havewhereNote that and thus is an orthogonal matrix. Next, we can use to define the new real representation.

Definition 6. For and which is unit and pure imaginary. We defineIt follows from Theorem 1 and Lemma 2 that the real representation can map a ϕ-Hermitian matrix or ϕ-skew-Hermitian matrix over into a skew-symmetric or symmetric matrix over . This will be a critical technique in tackling the problem of the paper.
DenotingThen, by direct calculation, we have the following lemma.

Lemma 3. are all orthogonal matrices, and

Next, we summarize the properties of real representations in the following proposition. Some properties of σ are given by [21, 22] and the others can be verified directly.

Proposition 1. Let . Then(a)(b)(c)(i) (ii) (d)(e)(f)If taking , we get real representations for η-(anti)-Hermitian matrices. Those real representations can map an η-Hermitian matrix or η-skew-Hermitian matrix over into a skew-symmetric or symmetric matrix over . They could be useful in the studying of η-(anti)-Hermitian matrix.

Definition 7. For , defineAs corollary of Proposition 1, we have:

Proposition 2. Let . Then(a)(b)(c)(d)(e)

4. ϕ-Hermitian or ϕ-Skew-Hermitian Solution to AX = B

In this section, we apply some new real representations given above to discuss the solvability conditions and ϕ-Hermitian or ϕ-skew-Hermitian solution of the quaternion matrix equation . We first begin with a useful lemma, which can be slightly modified from to .

Lemma 4 [23]. Let Then, the matrix equation has a solution if and only if and . In which case, the general solution to iswhere is an arbitrary matrix.

Applying Lemma 4 and real representations , we get the following statements.

Theorem 3. Let be given, where η is a unit and pure imaginary quaternion. Then, the quaternion matrix equationhas a ϕ-Hermitian (ϕ-skew-Hermitian) solution () if and only if one of the following equivalent statements hold:(a)The real matrix equation has a real solution ().Moreover, if (27) is consistent, thenis a solution to (27), wherewithand is arbitrary.

Proof. We only prove the case of . Suppose that X is a ϕ-Hermitian solution of (27), then it follows from (b) and (d) of Proposition 2 that and Postmultiplying the both sides with givesSince then (33) can be rewritten as Hence, is a skew-symmetric solution to (28).
Conversely, if (28) has a skew-symmetric solution i.e., Then, using (ii) in (c) of Proposition 1 givesNote that Hence,Then, by Lemma 3,which follows that are skew-symmetric solutions of (28). Then, so is For the given skew-symmetric solution Y, we setBy direct computationwhereNow, construct a quaternion matrixIt can be seen from (f) of Proposition 1 that is a skew-symmetric solution of (28); thus, and Put . η is unit and pure imaginary; thus, . Then, Observing thatThus, substituting it into gives As we recall that it is equivalent to Since . By Lemma 2 and (d) of Proposition 1, Therefore, X is a ϕ-Hermitian solution of (27). That is to say, once the skew-symmetric solution Y to (28) is given, then the blocks can be read directly from Y. Then, Z can be determined immediately by (40). Thus, our required quaternion solution X follows by taking Z into the formula By Lemma 4, Y given by (32) is a skew-symmetric solution to (28), and thus X in (30) is a ϕ-Hermitian solution to (27).
At the end, we can see that (27) has a solution if and only if its real matrix equation (28) has a skew-symmetric solution.

Remark 1. If we set , then ϕ-Hermitian or ϕ-skew-Hermitian matrix is reduced to the well-known η-Hermitian or η-anti-Hermitian matrix. Thus, the η-Hermitian or η-anti-Hermitian solution to (27) can be discussed by simply replacing by and by , respectively.

Example 5. Find a ϕ-skew-Hermitian solution to the quaternion matrix equation where By Theorem 3, its corresponding real matrix equation isSince and . Thus, the real equation has a symmetric solutionBy the formula of X in (30) in Theorem 3, the quaternion matrix equation also has a solutionwhich one can verify that .

5. Conclusion

In this paper, we give a complete characterization of the nonstandard involutions ϕ of and their conjugacy properties. Basing on the characterization of the nonstandard involutions, we present a new real representation of a quaternion matrix, which maps a ϕ-Hermitian or ϕ-skew-Hermitian quaternion matrix into a skew-symmetric or symmetric real matrix. By using this approach, we derive the necessary and sufficient conditions for the existence of a ϕ-Hermitian solution or ϕ-skew-Hermitian solution of the quaternion matrix equation (25). Furthermore, we get solutions of (25) when it is solvable. Moreover, Example 5 is presented to illustrate our results.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Disclosure

The first author gave a presentation of the abstract of this paper at “The 5th International Conference on Matrix Inequalities and Matrix Equations (MIME 2019)” that was held in Guilin, China, on June 7–9, 2019.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work of the first author was supported by the Science and Technology Development Fund, Macao SAR (grant no. 185/2017/A3), and National Natural Science Foundation for the Youth of China (grant no. 11701598). The work of the third author was supported by the National Natural Science Foundation of China (grant no. 11801354).