( Anti-) Hermitian Generalized ( Anti-) Hamiltonian Solution to a System of Matrix Equations

Wemainly solve three problems. Firstly, by the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonianmatrices, the necessary and sufficient conditions for the existence of and the expression for the (anti-)Hermitian generalized (anti-)Hamiltonian solutions to the system of matrix equations AX = B, XC = D are derived, respectively. Secondly, the optimal approximation solution min X∈K ‖X − X‖ is obtained, where K is the (anti-)Hermitian generalized (anti-)Hamiltonian solution set of the above system and X is the given matrix. Thirdly, the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solutions are considered. In addition, algorithms about computing the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solution and the corresponding numerical examples are presented.


Introduction
Throughout this paper, the set of all  ×  complex matrices, the set of all  ×  Hermitian matrices, the set of all  ×  anti-Hermitian matrices, the set of all  ×  unitary matrices, and the set of all  ×  antisymmetric orthogonal matrices are denoted, respectively, by C × , C × , C × , C × , and R × .The symbol   represents an identity matrix of order  and (),  † , and  * , respectively, stand for the rank, the Moore-Penrose inverse, and the conjugate transpose of matrix .For two matrices ,  ∈ C × , the inner product is defined by ⟨, ⟩ = tr( * ).Obviously, C × is a complete inner product space.The norm ‖ ⋅ ‖, induced by the inner product, is called the Frobenius norm. *  stands for the Hadamard product of two matrices  and .For  ∈ C × , two matrices   and   , respectively, represent two orthogonal projectors   =   −  †  and   =   −  † , both of which satisfy The Hamiltonian matrices defined as in [1] are very important in engineering (see [2] and the references therein).Moreover, using Hamiltonian matrices to solve algebraic matrix Riccati equation is a very effective method in optimal control theory [3][4][5].As the extension of the Hamiltonian matrices, the following four definitions, which can also be found in [1,6,7], are given.Without special statement, we in this paper always assume that  ∈ R 2×2 satisfies   = −,    =   =   .
( The well-known system of matrix equations with unknown matrix , has attracted much attention and has been widely and deeply studied by many authors.For example, Khatri and Mitra [8] in 1976 established the Hermitian and nonnegative definite solution to the system (3).Mitra [9] in 1984 gave the system (3) the minimal rank solution over the complex field C. Wang in [10] and Wang et al. [11], respectively, investigated the bisymmetric and centrosymmetric solutions over the quaternion algebra and obtained the bisymmetric nonnegative definite solutions with extremal ranks and inertias to the system (3).Xu in [12] studied the common Hermitian and positive solutions to the adjointable operator equations (3).Yuan in [13] presented the least squares solutions to the system (3).Some other results concerning the system (3) can be found in [14][15][16][17][18][19][20][21][22][23].
As special cases of the system (3), the classical matrix equations  =  and  =  have also been investigated (see, e.g., [1,2,[5][6][7][24][25][26][27][28][29][30][31]).For instance, Dai [24], by means of the singular value decomposition, derived the symmetric solution to equation  = .Guan and Jiang [6], using the decomposition of the anti-Hermitian generalized anti-Hamiltonian matrices, derived the least squares solution to equation  = .Zhang et al. in [29] and [1], respectively, obtained the general expression of the least squares Hermitian generalized Hamiltonian solutions to equation  =  and got the unite optimal approximation solution in the least squares solutions set and gave the solvable conditions and the general representation of the Hermitian generalized Hamiltonian solutions to equation  = , by using the singular value decomposition and the properties of Hermitian generalized Hamiltonian matrices.
As far as we know, there has been little information on studying the (anti-)Hermitian generalized (anti-) Hamiltonian solution to the system (3) over C 2×2 .So, motived by the work mentioned above, especially the work in [6,7,26,29,30], we, in this paper, are mainly concerned with the following three problems.
where  is the solution set of Problem 5.
The remainder of this paper is arranged as follows.In Section 2, some lemmas will be introduced, which will be useful for us to obtain the solutions to Problems 5-7.In Section 3, by applying the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices, the solvability condition and the explicit expression of the solution to Problem 5 will be derived.In Section 4, the optimal approximation solution to Problem 6 will be established.In Section 5, the solution to Problem 7 will be investigated and meanwhile the minimum norm of the solution will be obtained.In Section 6, algorithms and numerical examples about computing the solution to Problem 7 will be provided.Finally, in Section 7, some conclusions will be made.

Preliminaries
In this section, we focus on introducing some lemmas, which will play key roles in solving Problems 5-7.
Taking into account Definitions 1-4 and the eigenvalue decomposition of the matrix  ∈ R 2×2 , it is not difficult to conclude that the following decompositions of the (anti-) Hermitian generalized (anti-)Hamiltonian matrices hold, some of which can also be seen in [6,26,29,30].

Lemma 8. Let the eigenvalue decomposition of matrix 𝐽 ∈ 𝐴𝑆𝑂R
where  ∈ C in which case the general solutions can be expressed as where  ∈ C × is arbitrary.
By applying the singular value decomposition, similar to the proof of Theorem 1 in [24], the following lemma can be shown.
Lemma 13.Assume ,  ∈ C × .Let the singular value decomposition of  be where where Then the matrix equation has Hermitian solutions if and only if in which case the Hermitian solution can be expressed as where  22 ∈ C (−)×(−) is arbitrary.
By the similar way, the following lemma can also be verified.
Lemma 14. Assume ,  ∈ C × .Let the singular value decomposition of  be where where Then the matrix equation has an anti-Hermitian solution if and only if in which case the anti-Hermitian solution can be expressed as where  22 ∈ C (−)×(−) is arbitrary.
Lemma 15 (see [31]).Given   ,   ∈ C ×(+) ,   ,   ∈ C ×(+) , suppose that the matrices   and   , respectively, have the following singular value decompositions: where Then the solution set of the problem consists of matrices  12 ∈ C × with the following form: where and Lemma 16.Given ,  ∈ C × , let the singular value decomposition of , the partitions of  * and  *  be, respectively, as in ( 14)- (16).Then the least squares Hermitian solution to the matrix equation (18) can be expressed as where and  22 ∈ C (−)×(−) is arbitrary.
Proof.Combining ( 14)-( 16) and the unitary invariance of the Frobenius norm, it is easy to obtain that ‖ − ‖ Hence, there exists a unique solution  11 = (x  ) ∈ C × for (36) such that That is, where When  12 can be expressed as (37) gets its minimum.Therefore, the least squares Hermitian solution to (18) can be described as (33).
By the similar way, the following result can be obtained.
Lemma 17.Given ,  ∈ C × , let the singular value decomposition of , the partitions of  * , and  *  be, respectively, as in ( 21)- (23).Then the least squares anti-Hermitian solution to the matrix equation  =  can be expressed as where and  22 ∈ C (−)×(−) is arbitrary.
Lemma 18 (see [20]).Given  ∈ C × ,  ∈ C × , and  ∈ C × , then the matrix equation  =  has a solution if and only if in which case the general solution is where  ∈ C × is arbitrary.
Lemma 19.Let ,  ∈ C × .Then there exists a unique matrix where

The Solvability Conditions and the Expression of the Solution to Problem 5
In this section, our purpose is to derive the necessary and sufficient conditions of and the explicit expression of the solution to Problem 5 by using the results introduced in Section 2.
Then Problem 5 has a solution  ∈ C 2×2 if and only if in which case the Hermitian generalized Hamiltonian solution to Problem 5 can be expressed as where and  ∈ C × is arbitrary.
Proof.It follows from ( 7) and ( 49)-(52) that the system (3) can be transformed into the following system of matrix equations: Then, combining (53) and (54) yields that Thus, by Lemma 12, the system (59) has a solution  12 ∈ C × if and only if all equalities in (55) hold, in which case the solution can be written as (57).So the solution to system (3) can be expressed as (56).
Remark 21.Let  and  vanish in Theorem 20.Partition Then the matrix equation  =  has Hermitian generalized Hamiltonian solutions if and only if in which case its solution can be described as where and  ∈ C × is arbitrary.It is clear that this result is different from Theorem 3.1 given in [1].
Similarly, by Lemmas 9 and 12, we can get the anti-Hermitian generalized anti-Hamiltonian solution to system (3).Theorem 22.Given ,  ∈ C ×2 , ,  ∈ C 2× , let the decomposition of  ∈ C 2×2 be (8)., ,  * , and  * , respectively, have the partitions as in (49)-(52).Put Then Problem 5 has a solution  ∈ C 2×2 if and only if in which case the anti-Hermitian generalized anti-Hamiltonian solution to Problem 5 can be expressed as where and  ∈ C × is arbitrary.Now, we investigate the Hermitian generalized anti-Hamiltonian solution to the system (3).
Let the singular value decompositions of  and  be, respectively, where  ∈ C ) ; (75) where Then Problem 5 has a solution  ∈ C 2×2 if and only if in which case the Hermitian generalized anti-Hamiltonian solution to Problem 5 can be described as where and  22 ∈ C (−)×(−) , X22 ∈ C (−)×(−) are arbitrary.
Proof.It can be derived from ( 9), (49)-(52), and (68)-(69) that the system (3) is consistent if and only if the following two equations: From Lemmas 11 and 14, it is not difficult to obtain the anti-Hermitian generalized Hamiltonian solution to Problem 5, which can be described as follows.

The Expression of the Unique Solution to Problem 6
In this section, our aim is to derive the optimal approximation solution to Problem 6.
Theorem 25.Given X ∈ C 2×2 , under the hypotheses of Theorem 20, let If Problem 5 has Hermitian generalized Hamiltonian solutions, then Problem 6 has a unique solution X ∈ C 2×2 if and only if in which case the unique solution X can be expressed as where (96) Proof.When the Hermitian generalized Hamiltonian solution set  of Problem 5 is nonempty, it is not difficult to verify that  is a closed convex set.Then by [33], Problem 6 has a unique solution X ∈ C 2×2 .From Theorem 20, for any  ∈ ,  can be expressed as where where  ∈ C × is arbitrary.Inserting (102) into (97), and then combining (94) yields (95).
Analogously, the following theorem can be shown.
Theorem 26.Given X ∈ C 2×2 , under the hypotheses of Theorem 22, let If Problem 5 has anti-Hermitian generalized anti-Hamiltonian solutions, then Problem 6 has a unique solution X ∈ C 2×2 if and only if in which case the unique solution X can be expressed as where Now, we give the unique Hermitian generalized anti-Hamiltonian solution to Problem 6. ) , If Problem 5 has Hermitian generalized anti-Hamiltonian solutions, then the unique solution X ∈ C 2×2 to Problem 6 can be expressed as where Proof.When the Hermitian generalized anti-Hamiltonian solution set  of Problem 5 is nonempty, it is easy to prove that  is a closed convex set.Then, Problem 6 has a unique solution X ∈ C 2×2 by the aid of [33].For any  ∈ , due to Theorem 23,  can be expressed as where  11 and  22 have the expressions as in ( 81) and (82).
By the method used in Theorem 27, the following theorem can also be shown. where (121)

The Expression of the Solution to Problem 7
If the solvability conditions of linear matrix equations are not satisfied, the least squares solution is usually considered.So, in this section, the solution to Problem 7 is constructed.
By the same way, we can also derive the least squares anti-Hermitian generalized anti-Hamiltonian solution to Problem 7.
Proof.It follows from ( 9), ( 49)-( 52), ( 68 (2) Similarly, the algorithm about computing the least squares anti-Hermitian generalized anti-Hamiltonian solution to Problem 7 can be shown.We omit it here.Now, we provide another algorithm to compute the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7.
Step 2. Compute the eigenvalue decomposition of  according to (6).
Step 8. Compute  according to (9) (2) Similarly, the algorithm about computing the least squares anti-Hermitian generalized Hamiltonian solution to Problem 7 can be obtained.We also omit it here.

Conclusions
In the previous sections, using the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices, the necessary and sufficient conditions for the existence of and the expression for the solution to Problem 5 have been firstly derived, respectively.Then the solutions to Problems 6 and 7 have been individually given.Finally, algorithms have been given to compute the least squares Hermitian generalized Hamiltonian solution and the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7, and the corresponding examples have also been presented to show that the algorithms are reasonable.

) Definition 1 . 2 Mathematical
A matrix  ∈ C 2×2 is said to be a Hermitian generalized Hamiltonian matrix if  =  * and  =  * .Definition 2. A matrix  ∈ C 2×2 is said to be a Hermitian generalized anti-Hamiltonian matrix if  =  * and  = − * .Definition 3. A matrix  ∈ C 2×2 is said to be an anti-Hermitian generalized anti-Hamiltonian matrix if  = − * and  = − * .Problems in Engineering Definition 4. A matrix  ∈ C 2×2 is said to be an anti-Hermitian generalized Hamiltonian matrix if  = − * and  =  * .

Theorem 23 .
Given ,  ∈ C ×2 , ,  ∈ C 2× , let the decomposition of  ∈ C 2×2 be(9)., ,  * , and  * , respectively, have the partitions as in (49)-(52).Denote where  11 ,  22 ∈ C × are arbitrary.Given  ∈ C × ,  ∈ C × ,  ∈ C × , and  ∈ C × , then the system of matrix equations 2×2.Then  ∈ C 2×2 if and only if 11 such that (83) holds if and only if all equalities in (78) hold, in which case the solution can be written as (81).By the similar way, there exists Hermitian solution  22 such that (84) holds if and only if all equalities in (79) hold, in which case the solution can be described as (82).Therefore, the Hermitian generalized anti-Hamiltonian solution to Problem 5 can be expressed as (80).
(31)ralized anti-Hamiltonian solution with minimum norm to Problem 7 can be described as(8), where  12 has the expression as in(31)with   22 = 0.At present, we give the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7.
with  11 and  22 having the expressions as in (125) and (126), where  22 = 0, X22 = 0.At last, on the basis of Lemma 17, we can obtain the least squares anti-Hermitian generalized Hamiltonian solution to Problem 7, the proof of which is analogous to the proof of Theorem 33.Remark 39. (1) There exists a unique least squares Hermitian generalized Hamiltonian solution to Problem 7 if and only if both   and   in Theorem 29 have full row ranks.Example 38 just illustrates it.
, and output .Example 41.Let , , , ,  be as given in Example 38.It is not difficult to prove that the conditions in (78) and (79) do not hold.So, according to Algorithm 40, the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7 can be written as      −  *     = 0.0000, Remark 42. (1) There exists a unique least squares Hermitian generalized anti-Hamiltonian solution to Problem 7 if and only if both  and  in Theorem 33 have full column ranks.Example 41 is just the case.