Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 539215, 13 pages

http://dx.doi.org/10.1155/2014/539215

Research Article

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

Juan Yu,1,2 Qing-Wen Wang,¹ and Chang-Zhou Dong³

¹Department of Mathematics, Shanghai University, 99 Shangda Road, Shanghai 200444, China

²Department of Basic Mathematics, China University of Petroleum, Qingdao 266580, China

³School of Mathematics and Science, Shijiazhuang University of Economics, Shijiazhuang 050031, China

Received 12 August 2013; Revised 14 November 2013; Accepted 14 November 2013; Published 25 March 2014

Academic Editor: Masoud Hajarian

Copyright © 2014 Juan Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We mainly solve three problems. Firstly, by the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices, 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 are derived, respectively. Secondly, the optimal approximation solution is obtained, where is the (anti-)Hermitian generalized (anti-)Hamiltonian solution set of the above system and 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.

1. 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 , , and . 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 , the inner product is defined by . Obviously, 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 , 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 [3–5]. 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 satisfies

Definition 1. A matrix is said to be a Hermitian generalized Hamiltonian matrix if and .

Definition 2. A matrix is said to be a Hermitian generalized anti-Hamiltonian matrix if and .

Definition 3. A matrix is said to be an anti-Hermitian generalized anti-Hamiltonian matrix if and .

Definition 4. A matrix is said to be an anti-Hermitian generalized Hamiltonian matrix if and .

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 . 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 [14–23].

As special cases of the system (3), the classical matrix equations and have also been investigated (see, e.g., [1, 2, 5–7, 24–31]). 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 . 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.

Problem 5. Given ,, find , or such that the system (3) holds.

Problem 6. Given , find such that

where is the solution set of Problem 5.

Problem 7. Let ,. Find , or such that

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.

2. 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 , 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 be

where . Then if and only if can be expressed as

where are arbitrary.

Lemma 9. Let the eigenvalue decomposition of matrix be (6). Then if and only if can be expressed as

where is arbitrary.

Lemma 10. Let the eigenvalue decomposition of matrix be (6). Then if and only if can be expressed as

where are arbitrary.

Lemma 11. Let the eigenvalue decomposition of matrix be (6). Then if and only if can be expressed as

where are arbitrary.

Lemma 12 (see [20]). Given , and , then the system of matrix equations

has a solution if and only if

in which case the general solutions can be expressed as

where 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 . Let the singular value decomposition of be

where

Partition

where

Then the matrix equation

has Hermitian solutions if and only if

in which case the Hermitian solution can be expressed as

where is arbitrary.

By the similar way, the following lemma can also be verified.

Lemma 14. Assume . Let the singular value decomposition of be

where

Partition

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 is arbitrary.

Lemma 15 (see [31]). Given , , suppose that the matrices and , respectively, have the following singular value decompositions:

where

Then the solution set of the problem

consists of matrices with the following form:

where

and is arbitrary.

Lemma 16. Given , 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 is arbitrary.

Proof. Combining (14)–(16) and the unitary invariance of the Frobenius norm, it is easy to obtain that

Then reaches its minimum if and only if

reach their minimum. For , , since , then

Hence, there exists a unique solution for (36) such that

That is,

where

When 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 , 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 is arbitrary.

Lemma 18 (see [20]). Given , , and , then the matrix equation has a solution if and only if

in which case the general solution is

where is arbitrary.

The following lemma is due to [25, 32] or [29, Lemma  5].

Lemma 19. Let . Then there exists a unique matrix such that

where

3. 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.

Theorem 20. Given , let the decomposition of be (7). Partition

Then Problem 5 has a solution if and only if

in which case the Hermitian generalized Hamiltonian solution to Problem 5 can be expressed as

where

and 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 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 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 , , let the decomposition of be (8). , and , respectively, have the partitions as in (49)–(52). Put

Then Problem 5 has a solution if and only if

in which case the anti-Hermitian generalized anti-Hamiltonian solution to Problem 5 can be expressed as

where

and is arbitrary.

Now, we investigate the Hermitian generalized anti-Hamiltonian solution to the system (3).

Theorem 23. Given , , let the decomposition of be (9). , and , respectively, have the partitions as in (49)–(52). Denote

Let the singular value decompositions of and be, respectively,

where

Set

where

Then Problem 5 has a solution if and only if

in which case the Hermitian generalized anti-Hamiltonian solution to Problem 5 can be described as

where

and , 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:

are solvable. By (70), (73), and (74), and then combining Lemma 13, we can obtain that there exists Hermitian solution 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 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).

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.

Theorem 24. Given , , let the decomposition of be (10). , and , respectively, have the partitions as in (49)–(52). Denote

Let the singular value decompositions of and be, respectively,

where

Set

where

Then Problem 5 has a solution if and only if

in which case the anti-Hermitian generalized Hamiltonian solution to Problem 5 can be described as

where

and , are arbitrary.

4. 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 , under the hypotheses of Theorem 20, let

If Problem 5 has Hermitian generalized Hamiltonian solutions, then Problem 6 has a unique solution if and only if

in which case the unique solution can be expressed as

where

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 . From Theorem 20, for any , can be expressed as

where

and is arbitrary. Then it follows from the equalities in (93) and (97) and the unitary invariance of the Frobenius norm that

Thus, Problem 6 has a unique solution if and only if there exists such that

reaches its minimum. Therefore, by Lemma 19, (100) arrives at its minimum if and only if there exists such that the matrix equation

holds, which, by Lemma 18, has a solution if and only if (94) holds, in which case the solution can be expressed as

where is arbitrary. Inserting (102) into (97), and then combining (94) yields (95).

Analogously, the following theorem can be shown.

Theorem 26. Given , under the hypotheses of Theorem 22, let

If Problem 5 has anti-Hermitian generalized anti-Hamiltonian solutions, then Problem 6 has a unique solution if and only if

in which case the unique solution can be expressed as

where

Now, we give the unique Hermitian generalized anti-Hamiltonian solution to Problem 6.

Theorem 27. Given , under the hypotheses of Theorem 23, let

If Problem 5 has Hermitian generalized anti-Hamiltonian solutions, then the unique solution 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 by the aid of [33]. For any , due to Theorem 23, can be expressed as

where and have the expressions as in (81) and (82). Combining the equalities in (80)–(82) and (107) and the unitary invariance of the Frobenius norm yields that

So,

By (81), (108), and the unitary invariance of the Frobenius norm, we obtain

Then

Therefore, when can be expressed as

holds. Then combining (81) yields (111). Similarly, we can derive the expression in (112) by (82) and (109). Thus, (110) is the unique solution to Problem 6.

By the method used in Theorem 27, the following theorem can also be shown.

Theorem 28. Given , under the hypotheses of Theorem 24, let

If Problem 5 has anti-Hermitian generalized Hamiltonian solutions, then the unique solution to Problem 6 can be expressed as

where

5. 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.

Theorem 29. Given , , let the decomposition of be (7). , and , respectively, have the partitions as in (49)–(52) and (54). Denote

Let the singular value decompositions of and be as given in (28). Then the least squares Hermitian generalized Hamiltonian solution to Problem 7 can be described as (7), where has the expression as in (31).

Proof. Combining (7), (49)–(52), (54), (122), and the unitary invariance of the Frobenius norm yields that

Therefore, by Lemma 15, if has the expression as in (31), then (123) reaches its minimum. Then, substituting (31) into (7), we obtain the least squares Hermitian generalized Hamiltonian solution to Problem 7.

Corollary 30. Given , , under the conditions of Theorem 29, the least squares Hermitian generalized Hamiltonian solution with minimum norm to Problem 7 can be described as (7), where has the expression as in (31) with .

By the same way, we can also derive the least squares anti-Hermitian generalized anti-Hamiltonian solution to Problem 7.

Theorem 31. Given , , let the decomposition of be (8). , and , respectively, have the partitions as in (49)–(52). Denote

Let the singular value decompositions of and be as in (28). Then the least squares anti-Hermitian generalized anti-Hamiltonian solution to Problem 7 can be described as (8), where has the expression as in (31).

Corollary 32. Given , , under the conditions of Theorem 31, the least squares anti-Hermitian generalized anti-Hamiltonian solution with minimum norm to Problem 7 can be described as (8), where has the expression as in (31) with .

At present, we give the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7.

Theorem 33. Assume , . Let the decomposition of be (9). , , , and , respectively, have the partitions as in (49)–(52), (68), and (69). Let the singular value decompositions of and be, respectively, (70) and (71), , have the partitions as in (73)–(76). Then the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7 can be expressed as (9) with

where

and , are arbitrary.

Proof. It follows from (9), (49)–(52), (68), (69), and the unitary invariance of the Frobenius norm that

Then

gains its minimum value if and only if

So, by (68), (70), (73), and (74) and then combining Lemma 16, we get that if has the expression as in (125), then (130) holds. Similarly, if has the expression as in (126), then (131) holds. Thus, the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7 can be expressed as (9), where and have the expressions as in (125) and (126).

Corollary 34. Given , , under the conditions of Theorem 33, the least squares Hermitian generalized anti-Hamiltonian solution with minimum norm to Problem 7 can be expressed as (9) with and having the expressions as in (125) and (126), where .

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.

Theorem 35. Given , , let the decomposition of be (10). , and , respectively, have the partitions as in (49)–(52), (85). Assume that the singular value decompositions of and are, respectively, expressed as in (86) and , have the partitions as in (88). Then the least squares anti-Hermitian generalized Hamiltonian solution to Problem 7 can be expressed as (10) with

where

and , are arbitrary.

Corollary 36. Given , , under the conditions of Theorem 35, the least squares anti-Hermitian generalized Hamiltonian solution with minimum norm to Problem 7 can be expressed as (10) with and having the expressions as in (132), where .

6. Algorithms and Numerical Examples

In this section, algorithms are given to compute the solution to Problem 7, and meanwhile some numerical examples are presented to show that the algorithms provided are feasible. Note that all the tests are performed by MATLAB 7.6.

An algorithm is firstly presented to compute the least squares Hermitian generalized Hamiltonian solution to Problem 7.

Algorithm 37. Step  1. Input .

Step  2. Compute the eigenvalue decomposition of according to (6).

Step  3. Compute according to (49)–(52).

Step  4. Compute according to (53) and (54). If the conditions in (55) hold, then compute the Hermitian generalized Hamiltonian solution to Problem 5 according to (56) and (57). Otherwise, turn to Step  5.

Step  5. Compute according to (53) and (122).

Step  6. Compute the singular value decompositions of and according to (28).

Step  7. Compute according to (31).

Step  8. Compute according to (7), and output .

Example 38. Given

it can be easily verified that the conditions in (55) are not satisfied. Then, according to Algorithm 37, the least squares Hermitian generalized Hamiltonian solution to Problem 7 can be expressed as

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.

(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.

Algorithm 40. Step  1. Input .

Step  2. Compute the eigenvalue decomposition of according to (6).

Step  3. Compute according to (49)–(52).

Step  4. Compute according to (68) and (69).

Step  5. Compute the singular value decompositions of and according to (70)-(71).

Step  6. Compute the partitions of according to (74) and (76). If the conditions in (78) and (79) are all satisfied, then compute the Hermitian generalized anti-Hamiltonian solution to Problem 5 according to (80)–(82). Otherwise, turn to Step  7.

Step  7. Compute and according to (125) and (126).

Step  8. Compute according to (9), 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

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.

(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.

7. 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.

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 (11171205), the National Natural Science Foundation of China (11301330), the Key Project of Scientific Research Innovation Foundation of Shanghai Municipal Education Commission (13ZZ080), the Natural Science Foundation of Shanghai (11ZR1412500), the Youth Funds of Natural Science Foundation of Hebei province (A2012403013), and the Natural Science Foundation of Hebei province (A2012205028).

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