The Real and Complex Hermitian Solutions to a System of Quaternion Matrix Equations with Applications

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 equationsA1X C1, XB1 C2, and A3XA3 C3. Moreover, formulas of the maximal andminimal ranks of four real matrices X1, X2, X3, and X4 in solution X X1 X2i X3j X4k to the system mentioned above are derived. As applications, we give necessary and sufficient conditions for the quaternion matrix equations A1X C1, XB1 C2, A3XA3 C3, and A4XA ∗ 4 C4 to have real and complex Hermitian solutions.


Introduction
Throughout this paper, we denote the real number field by R; the complex field by C; the set of all m × n matrices over the quaternion algebra 1 by H m×n ; the identity matrix with the appropriate size by I; the transpose, the conjugate transpose, the column right space, the row left space of a matrix A over H by A T , A * , R A , N A , respectively; the dimension of R A by dim R A .By 1 , for a quaternion matrix A, dim R A dim N A .dim R A is called the rank of a quaternion matrix A and denoted by r A .The Moore-Penrose inverse of matrix A over H by A † which satisfies four Penrose equations AA † A A, A † AA † A † , AA † * AA † , and A † A * A † A. In this case, A † is unique and A † * A * † .Moreover, R A and L A stand for the two projectors L A I − A † A,

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and R A I − AA † induced by A. Clearly, R A and L A are idempotent and satisfies R A * R A , L A * L A , R A L A * , and R A * L A .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 AX B, AXB C and over the complex field C, 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 C in terms of generalized inverses, respectively.In 10 , Tian and Liu established the solvability conditions for to have a common Hermitian solution over C 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 1.6 for operators between Hilbert C * -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 X 0 and X 1 in solution X X 0 iX 1 to AXB C over C with its applications.Liu et al. 9 derived the maximal and minimal ranks of the two real matrices X 0 and X 1 in a Hermitian solution X X 0 iX 1 of 1.3 , where B * B. 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 AXB C, recently 24-27 .To our knowledge, the necessary and sufficient conditions for 1.5 over H 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 H and its applications.This paper is organized as follows.In Section 2, we first derive formulas of extremal ranks of four real matrices X 1 , X 2 , X 3 , and X 4 in quaternion solution X X 1 X 2 i X 3 j X 4 k to 1.5 over H, then give necessary and sufficient conditions for 1.5 over H 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 H to have real and complex solutions.

The Real and Complex Hermitian Solutions to System 1.5 Over H
In this section, we first give a solvability condition and an expression of the general Hermitian solution to 1.5 over H, then consider the maximal and minimal ranks of four real matrices X 1 , X 2 , X 3 , and X 4 in solution X X 1 X 2 i X 3 j X 4 k to 1.5 over H, last, investigate the real and complex Hermitian solutions to 1.5 over H.
For an arbitrary matrix M t M t1 M t2 i M t3 j M t4 k ∈ H m×n where M t1 , M t2 , M t3 , and M t4 are real matrices, we define a map φ • from H m×n to R 4m×4n by 2.1 By 2.1 , it is easy to verify that φ • satisfies the following properties.
The following lemmas provide us with some useful results over C, which can be generalized to H.

2.3
In that case, the general Hermitian solution of 1.3 can be expressed as where

2.5
In that case, the general Hermitian solution of 1.5 can be expressed as where

2.7
where , and C 4 ∈ H q 4 ×n be given.Then the matrix equation 11

has a Hermitian solution over H if and only if the system of matrix equations
International Journal of Mathematics and Mathematical Sciences has a symmetric solution over R. In that case, the general Hermitian solution of 1.5 over H can be written as where

34
Y 43 are the general solutions of 2.13 over R. Written in an explicit form, X 1 , X 2 , X 3 , and X 4 in 2.14 are International Journal of Mathematics and Mathematical Sciences 7 2.17 where P 1 I n , 0, 0, 0 , P 2 0, I n , 0, 0 , P 3 0, 0, I n , 0 , P 4 0, 0, 0, I n , 2.19 φ X 0 is a particular symmetric solution to 2.13 , and V is arbitrary real matrices with compatible sizes.
Proof.Suppose that 1.5 has a Hermitian solution X over H. Applying properties a and b of φ • to 1.5 yields implying that φ X is a real symmetric solution to 2.13 .Conversely, suppose that 2.13 has a real symmetric solution

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That is, then by property d of φ • ,

2.23
Hence, , and S n Y S −1 n are also symmetric solutions of 2.13 .Thus, is a symmetric solution of 2.13 , where

2.27
Then by 2.1 , Hence, by the property a , we know that X is a Hermitian solution of 1.5 .Observe that Y ij , i, j 1, 2, 3, 4, in 2.13 can be written as

2.29
From Lemma 2.2, the general Hermitian solution to 2.13 can be written as where V ∈ R is arbitrary.Hence,
Now we consider the maximal and minimal ranks of four real matrices X 1 , X 2 , X 3 , and X 4 in solution X X 1 X 2 i X 3 j X 4 k to 1.5 over H.

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Theorem 2.7.Suppose that system 1.5 over H has a Hermitian solution, and

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Then the maximal and minimal ranks of X i , i 1, 2, 3, 4, in Hermitian solution X X 1 X 2 i X 3 j X 4 k to 1.5 are given by max where

2.35
Proof.We only prove the case that i 1.Similarly, we can get the results that i 2, 3, 4. Let note that L M is Hermtian; then L φ M is symmetric; hence 2.15 can be written as

2.37
Note that A A * and R B ⊆ R C * ; applying 2.9 and 2.10 to 2.37 yields max

2.40
Note that φ X 0 is a particular solution to 2.13 , it is not difficult to find by Lemma 2.3, block Gaussian elimination, and property e of φ • that

2.41
Note that International Journal of Mathematics and Mathematical Sciences 13

2.42
Similarly, we can obtain hold when i 2, 3, 4. In that case, the real solution of 1.5 can be expressed as X X 1 in 2.15 .b System 1.5 has a complex solution if and only if 2.44 hold when i 3, 4 or i 2, 4 or i 2, 3.In that case, the complex solutions of 1.5 can be expressed as X X 1 X 2 i or X X 1 X 3 j International Journal of Mathematics and Mathematical Sciences or X X 1 X 4 k, where X 1 , X 2 , X 3 , and X 4 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 X i 0, i 1, 2, 3, 4. Thus we can get the results of this Corollary.

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where Proof.From Corollary 2.8, system 1.5 over H has a real Hermitian solution if and only if 2.44 hold when i 2, 3, 4. By 2.15 , the real Hermitian solutions of 1.5 over H can be expressed as where φ Y 0 is a particular solution to φ A 4 Y ij 4 × 4 φ T A 4 φ C 4 and U is arbitrary matrices with compatible sizes.The expression of Y 1 can also be obtained from Lemma 2.1.Let P 1 , P 2 , P 3 , P 4 P,

3.7
Equating X 1 and Y 1 , we obtain the following equation: It is obvious that system 1.

3.9
We know by Lemma 2.5 that 3.9 is solvable if and only if the following four rank equalities hold

3.10
Under the conditions that the system 1.5 and the matrix equation A 4 Y A * 4 C 4 over H have Hermitian solutions, it is not difficult to show by Lemma 2.3 and block Gaussian elimination that 3.10 are equivalent to the four rank equalities 3.2 and 3.3 , respectively.Note that the processes are too much tedious; we omit them here.Obviously, the system 1.5 and the matrix equation A 4 Y A * 4 C 4 over H have a common real Hermitian solution if and only if 3.2 and 3.3 hold.Thus, the system 1.6 over H has a real Hermitian solution if and only if 2.44 hold when i 2, 3, 4, and 3.1 -3.3 hold.
Similarly, from Corollary 2.8, we know that the system 1.5 over H has a complex Hermitian solution if and only if 2.44 hold when i 3, 4, i 2, 4, or; i 2, 3; its complex Hermitian solutions can be expressed as X X 1 X 2 i, X X 1 X 3 j, or X X 1 X 4 k.It is also easy to derive the necessary and sufficient condition for the matrix equation A 4 Y A * 4 C 4 over H to have a complex Hermitian solution; its complex Hermitian solution can be expressed as By equating X 1 and Y 1 , X 2 and Y 2 , X 3 , and Y 3 , X 4 and Y 4 , respectively, we can derive the necessary and sufficient conditions for the system 1.6 over H to have a complex Hermitian solution.

Lemma 2.1 see
2, Lemma 2.1 .Let A ∈ H m×n , B B * ∈ H m×m be known, X ∈ H n×n unknown; then the system 1.3 has a Hermitian solution if and only if AA † B B.
is arbitrary matrix over H with compatible size.Let A 1 , C 1 ∈ H m×n ; B 1 , C 2 ∈ H n×s ; A 3 ∈ H r×n ; C 3 ∈ H r×r be known, X ∈ H n×n unknown, and

Lemma 2.3 see 21
is arbitrary matrix over H with compatible size., Lemma 2.4 .Let A ∈ H m×n , B ∈ H m×k , C ∈ H l×n , D ∈ H j×k , and E ∈ H l×i .Then they satisfy the following rank equalities.
3 plays an important role in simplifying ranks of various block matrices.

Real and Complex Hermitian Solutions to
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 H to have real and complex Hermitian solutions.Let A 1 , A 3 , B 1 , C 1 , C 2 ,and C 3 be defined in Lemma 2.2, A 4 ∈ H l×n , C 4 ∈ H l×l , and suppose that system 1.5 and the matrix equation A 4 Y A * 4 C 4 over H have Hermitian solutions X and Y ∈ H n×n , respectively.Then system 1.6 over H has a real Hermitian solution if and only if 2.44 hold when i 2, 3, 4, and 5where V is arbitrary matrices with compatible sizes.LetA 1 , C 1 0; B 1 , C 2 0; A 3 A 4 ; C 3 C 4 inCorollary 2.8 and 2.15 .It is easy to verify that the matrix equation A 4 Y A * 4 C 4 over H has a real Hermitian solution if and only if 3.1 hold and the real Hermitian solution can be expressed as 5 and the matrix equation A 4 Y A * 4 C 4 over H have common real Hermitian solution if and only if min r X 1 − Y 1 0, that is, X 1 − Y 1 0. Hence, we have the matrix equation