#### Abstract

We establish necessary and sufficient conditions for the existence of and the expressions for the general real and complex Hermitian solutions to the classical system of quaternion matrix equations . Moreover, formulas of the maximal and minimal ranks of four real matrices , and in solution to the system mentioned above are derived. As applications, we give necessary and sufficient conditions for the quaternion matrix equations , and to have real and complex Hermitian solutions.

#### 1. Introduction

Throughout this paper, we denote the real number field by ; the complex field by ; the set of all matrices over the quaternion algebra by ; the identity matrix with the appropriate size by ; the transpose, the conjugate transpose, the column right space, the row left space of a matrix over by , , respectively; the dimension of by . By [1], for a quaternion matrix . is called the rank of a quaternion matrix and denoted by . The Moore-Penrose inverse of matrix over by which satisfies four Penrose equations , and . In this case, is unique and . Moreover, and stand for the two projectors , and induced by . Clearly, and are idempotent and satisfies , and .

Hermitian solutions to some matrix equations were investigated by many authors. In 1976, Khatri and Mitra [2] gave necessary and sufficient conditions for the existence of the Hermitian solutions to the matrix equations and over the complex field , and presented explicit expressions for the general Hermitian solutions to them by generalized inverses when the solvability conditions were satisfied. Matrix equation that has symmetric patterns with Hermitian solutions appears in some application areas, such as vibration theory, statistics, and optimal control theory ([3–7]). Groß in [8], and Liu et al. in [9] gave the solvability conditions for Hermitian solution and its expressions of over in terms of generalized inverses, respectively. In [10], Tian and Liu established the solvability conditions for to have a common Hermitian solution over by the ranks of coefficient matrices. In [11], Tian derived the general common Hermitian solution of (1.4). Wang and Wu in [12] gave some necessary and sufficient conditions for the existence of the common Hermitian solution to equations for operators between Hilbert -modules by generalized inverses and range inclusion of matrices.

As is known to us, extremal ranks of some matrix expressions can be used to characterize nonsingularity, rank invariance, range inclusion of the corresponding matrix expressions, as well as solvability conditions of matrix equations ([4, 7, 9–24]). Real matrices and its extremal ranks in solutions to some complex matrix equation have been investigated by Tian and Liu ([9, 13–15]). Tian [13] gave the maximal and minimal ranks of two real matrices and in solution to over with its applications. Liu et al. [9] derived the maximal and minimal ranks of the two real matrices and in a Hermitian solution of (1.3), where . In order to investigate the real and complex solutions to quaternion matrix equations, Wang and his partners have been studying the real matrices in solutions to some quaternion matrix equations such as , recently ([24–27]). To our knowledge, the necessary and sufficient conditions for (1.5) over to have the real and complex Hermitian solutions have not been given so far. Motivated by the work mentioned above, we in this paper investigate the real and complex Hermitian solutions to system (1.5) over and its applications.

This paper is organized as follows. In Section 2, we first derive formulas of extremal ranks of four real matrices , and in quaternion solution to (1.5) over , then give necessary and sufficient conditions for (1.5) over to have real and complex solutions as well as the expressions of the real and complex solutions. As applications, we in Section 3 establish necessary and sufficient conditions for (1.6) over to have real and complex solutions.

#### 2. The Real and Complex Hermitian Solutions to System (1.5) Over

In this section, we first give a solvability condition and an expression of the general Hermitian solution to (1.5) over , then consider the maximal and minimal ranks of four real matrices , and in solution to (1.5) over , last, investigate the real and complex Hermitian solutions to (1.5) over .

For an arbitrary matrix where , and are real matrices, we define a map from to by By (2.1), it is easy to verify that satisfies the following properties.(a). (b).(c), . (d), where , (e). (f), .

The following lemmas provide us with some useful results over , which can be generalized to .

Lemma 2.1 (see [2, Lemma 2.1]). *Let be known, unknown; then the system (1.3) has a Hermitian solution if and only if
**
In that case, the general Hermitian solution of (1.3) can be expressed as
**
where is arbitrary matrix over with compatible size.*

Lemma 2.2 (see [12, Corollary 3.4]). *Let be known, unknown, and ; then the system (1.5) have a Hermitian solution if and only if
**
In that case, the general Hermitian solution of (1.5) can be expressed as
**
where is arbitrary matrix over with compatible size.*

Lemma 2.3 (see [21, Lemma 2.4]). *Let , and . Then they satisfy the following rank equalities.*(a)*. *(b)*. *(c)*. *(d)*. *

Lemma 2.3 plays an important role in simplifying ranks of various block matrices.

Lemma 2.4 (see [11, Theorem 4.1, Corollary 4.2]). *Let , and be given; then
**
where
**
If ,
*

Lemma 2.5 (see [28, Theorem 3.1]). *Let , and be given. Then the matrix equation
**
is consistent if and only if
*

Theorem 2.6. *System (1.5) has a Hermitian solution over if and only if the system of matrix equations
**
has a symmetric solution over . In that case, the general Hermitian solution of (1.5) over can be written as
**
where are the general solutions of (2.13) over . Written in an explicit form, , and in (2.14) are
**
where
** is a particular symmetric solution to (2.13), and is arbitrary real matrices with compatible sizes.*

*Proof. *Suppose that (1.5) has a Hermitian solution over . Applying properties (a) and (b) of to (1.5) yields
implying that is a real symmetric solution to (2.13).

Conversely, suppose that (2.13) has a real symmetric solution
That is,
then by property of ,
Hence,
implying that , and are also symmetric solutions of (2.13). Thus,
is a symmetric solution of (2.13), where
Let
Then by (2.1),
Hence, by the property (a), we know that is a Hermitian solution of (1.5). Observe that , in (2.13) can be written as
From Lemma 2.2, the general Hermitian solution to (2.13) can be written as
where is arbitrary. Hence,
where , substituting them into (2.14), yields the four real matrices , and in (2.15)–(2.18).

Now we consider the maximal and minimal ranks of four real matrices , and in solution to (1.5) over .

Theorem 2.7. *Suppose that system (1.5) over has a Hermitian solution, and **
Then the maximal and minimal ranks of , in Hermitian solution to (1.5) are given by**
where
*

*Proof. *We only prove the case that . Similarly, we can get the results that . Let
note that is Hermtian; then is symmetric; hence (2.15) can be written as
Note that and ; applying (2.9) and (2.10) to (2.37) yields
Let
Note that is a particular solution to (2.13), it is not difficult to find by Lemma 2.3, block Gaussian elimination, and property of that
Note that , then ; hence
Similarly, we can obtain
Substituting (2.41) and (2.43) into (2.38) and (2.39) yields (2.33) and (2.34), that is .

Corollary 2.8. *Suppose system (1.5) over have a Hermitian solution. Then we have the following.**
(a) System (1.5) has a real hermtian solution if and only if
**
hold when . In that case, the real solution of (1.5) can be expressed as in (2.15).**
(b) System (1.5) has a complex solution if and only if (2.44) hold when or or . In that case, the complex solutions of (1.5) can be expressed as or or , where , and are expressed as (2.15), (2.16), (2.17), and (2.18), respectively.*

*Proof. *From (2.34) we can get the necessary and sufficient conditions for . Thus we can get the results of this Corollary.

#### 3. Solvability Conditions for Real and Complex Hermitian Solutions to (1.6) Over

In this section, using the results of Theorem 2.6, Theorem 2.7, and Corollary 2.8, we give necessary and sufficient conditions for (1.6) over to have real and complex Hermitian solutions.

Theorem 3.1. *Let , and be defined in Lemma 2.2, , and suppose that system (1.5) and the matrix equation over have Hermitian solutions and , respectively. Then system (1.6) over has a real Hermitian solution if and only if (2.44) hold when , and
**
where
*

*Proof. *From Corollary 2.8, system (1.5) over has a real Hermitian solution if and only if (2.44) hold when . By (2.15), the real Hermitian solutions of (1.5) over can be expressed as
where is arbitrary matrices with compatible sizes.

Let ; in Corollary 2.8 and (2.15). It is easy to verify that the matrix equation over has a real Hermitian solution if and only if (3.1) hold and the real Hermitian solution can be expressed as
where is a particular solution to and is arbitrary matrices with compatible sizes. The expression of can also be obtained from Lemma 2.1. Let
Equating and , we obtain the following equation: