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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 514984, 9 pages
http://dx.doi.org/10.1155/2013/514984
Research Article

Ranks of a Constrained Hermitian Matrix Expression with Applications

Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

Received 12 November 2012; Accepted 4 January 2013

Academic Editor: Yang Zhang

Copyright © 2013 Shao-Wen Yu. 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 establish the formulas of the maximal and minimal ranks of the quaternion Hermitian matrix expression where is a Hermitian solution to quaternion matrix equations , , and . As applications, we give a new necessary and sufficient condition for the existence of Hermitian solution to the system of matrix equations , , , and , which was investigated by Wang and Wu, 2010, by rank equalities. In addition, extremal ranks of the generalized Hermitian Schur complement with respect to a Hermitian g-inverse of , which is a common solution to quaternion matrix equations and , are also considered.

1. Introduction

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

Mitra [2] investigated the system of matrix equations

Khatri and Mitra [3] gave necessary and sufficient conditions for the existence of the common Hermitian solution to (2) and presented an explicit expression for the general Hermitian solution to (2) by generalized inverses. Using the singular value decomposition (SVD), Yuan [4] investigated the general symmetric solution of (2) over the real number field . By the SVD, Dai and Lancaster [5] considered the symmetric solution of equation over , which was motivated and illustrated with an inverse problem of vibration theory. Groß [6], Tian and Liu [7] gave the solvability conditions for Hermitian solution and its expressions of (3) over in terms of generalized inverses, respectively. Liu, Tian and Takane [8] investigated ranks of Hermitian and skew-Hermitian solutions to the matrix equation (3). By using the generalized SVD, Chang and Wang [9] examined the symmetric solution to the matrix equations over . Note that all the matrix equations mentioned above are special cases of Wang and Wu [10] gave some necessary and sufficient conditions for the existence of the common Hermitian solution to (5) for operators between Hilbert C*-modules by generalized inverses and range inclusion of matrices. In view of the complicated computations of the generalized inverses of matrices, we naturally hope to establish a more practical, necessary, and sufficient condition for system (5) over quaternion algebra to have Hermitian solution by rank equalities.

As is known to us, solutions to matrix equations and ranks of solutions to matrix equations have been considered previously by many authors [1034], and extremal ranks of matrix expressions can be used to characterize their rank invariance, nonsingularity, range inclusion, and solvability conditions of matrix equations. Tian and Cheng [35] investigated the maximal and minimal ranks of with respect to with applications; Tian [36] gave the maximal and minimal ranks of subject to a consistent matrix equation Tian and Liu [7] established the solvability conditions for (4) to have a Hermitian solution over by the ranks of coefficient matrices. Wang and Jiang [20] derived extreme ranks of (skew)Hermitian solutions to a quaternion matrix equation . Wang, Yu and Lin [31] derived the extremal ranks of subject to a consistent system of matrix equations over and gave a new solvability condition to system

In matrix theory and its applications, there are many matrix expressions that have symmetric patterns or involve Hermitian (skew-Hermitian) matrices. For example, where , and are given and and are variable matrices. In recent papers [7, 8, 37, 38], Liu and Tian considered some maximization and minimization problems on the ranks of Hermitian matrix expressions (8).

Define a Hermitian matrix expression where ; we have an observation that by investigating extremal ranks of (9), where is a Hermitian solution to a system of matrix equations A new necessary and sufficient condition for system (5) to have Hermitian solution can be given by rank equalities, which is more practical than one given by generalized inverses and range inclusion of matrices.

It is well known that Schur complement is one of the most important matrix expressions in matrix theory; there have been many results in the literature on Schur complements and their applications [3941]. Tian [36, 42] has investigated the maximal and minimal ranks of Schur complements with applications.

Motivated by the work mentioned above, we in this paper investigate the extremal ranks of the quaternion Hermitian matrix expression (9) subject to the consistent system of quaternion matrix equations (10) and its applications. In Section 2, we derive the formulas of extremal ranks of (9) with respect to Hermitian solution of (10). As applications, in Section 3, we give a new, necessary, and sufficient condition for the existence of Hermitian solution to system (5) by rank equalities. In Section 4, we derive extremal ranks of generalized Hermitian Schur complement subject to (2). We also consider the rank invariance problem in Section 5.

2. Extremal Ranks of (9) Subject to System (10)

Corollary 8 in [10] over Hilbert C*-modules can be changed into the following lemma over .

Lemma 1. Let , be given, and ; then the following statements are equivalent:(1)the system (10) have a Hermitian solution,(2), (3); the equalities in (11) hold and
In that case, the general Hermitian solution of (10) can be expressed as where is Hermitian matrix over with compatible size.

Lemma 2 (see Lemma 2.4 in [24]). Let , , and . Then the following rank equalities hold:(a), (b), (c), (d).

Lemma 2 plays an important role in simplifying ranks of various block matrices.

Liu and Tian [38] has given the following lemma over a field. The result can be generalized to .

Lemma 3. Let , , and be given; then where
If ,

Now we consider the extremal ranks of the matrix expression (9) subject to the consistent system (10).

Theorem 4. Let be defined as Lemma 1, . Then the extremal ranks of the quaternion matrix expression defined as (9) subject to system (10) are the following: where

Proof. By Lemma 1, the general Hermitian solution of the system (10) can be expressed as where is Hermitian matrix over with appropriate size. Substituting (21) into (9) yields Put then
Note that and . Thus, applying (17) to (24), we get the following:
Now we simplify the ranks of block matrices in (25).
In view of Lemma 2, block Gaussian elimination, (11), (12), and (23), we have the following:
Substituting (26) into (25) yields (18) and (20).

In Theorem 4, letting vanish and be with appropriate size, respectively, we have the following.

Corollary 5. Assume that , , are given; then the maximal and minimal ranks of the Hermitian solution to the system (10) can be expressed as where

In Theorem 4, assuming that , and vanish, we have the following.

Corollary 6. Suppose that the matrix equation is consistent; then the extremal ranks of the quaternion matrix expression defined as (9) subject to are the following:

3. A Practical Solvability Condition for Hermitian Solution to System (5)

In this section, we use Theorem 4 to give a necessary and sufficient condition for the existence of Hermitian solution to system (5) by rank equalities.

Theorem 7. Let , , , , and be given; then the system (5) have Hermitian solution if and only if, (11), (13) hold, and the following equalities are all satisfied:

Proof. It is obvious that the system (5) have Hermitian solution if and only if the system (10) have Hermitian solution and where is defined as (9) subject to system (10). Let be a Hermitian solution to the system (5); then is a Hermitian solution to system (10) and satisfies . Hence, Lemma 1 yields , (11), (13), and (30). It follows from that (32) holds. Similarly, we can obtain (31).
Conversely, assume that , (11), (13) hold; then by Lemma 1, system (10) have Hermitian solution. By (20), (31)-(32), and we can get However,
Hence (33) holds, implying that the system (5) have Hermitian solution.

By Theorem 7, we can also get the following.

Corollary 8. Suppose that , , , and are those in Theorem 7; then the quaternion matrix equations and have common Hermitian solution if and only if (30) hold and the following equalities are satisfied:

Corollary 9. Suppose that , , , and , are Hermitian. Then and have a common Hermitian g-inverse which is a solution to the system (2) if and only if (11) holds and the following equalities are all satisfied:

4. Extremal Ranks of Schur Complement Subject to (2)

As is well known, for a given block matrix where and are Hermitian quaternion matrices with appropriate sizes, then the Hermitian Schur complement of in is defined as where is a Hermitian g-inverse of , that is, .

Now we use Theorem 4 to establish the extremal ranks of given by (42) with respect to which is a solution to system (2).

Theorem 10. Suppose , , are given and system (2) is consistent; then the extreme ranks of given by (42) with respect to which is a solution of (2) are the following: where

Proof. It is obvious that
Thus in Theorem 4 and its proof, letting , , we can easily get the proof.

In Theorem 10, let vanish. Then we can easily get the following.

Corollary 11. The extreme ranks of given by (42) with respect to are the following:

5. The Rank Invariance of (9)

As another application of Theorem 4, we in this section consider the rank invariance of the matrix expression (9) with respect to the Hermitian solution of system (10).

Theorem 12. Suppose that (10) have Hermitian solution; then the rank of defined by (9) with respect to the Hermitian solution of (10) is invariant if and only if or

Proof. It is obvious that the rank of with respect to Hermitian solution of system (10) is invariant if and only if
By (49), Theorem 4, and simplifications, we can get (47) and (48).

Corollary 13. The rank of defined by (42) with respect to which is a solution to system (2) is invariant if and only if or

Acknowledgments

This research was supported by the National Natural Science Foundation of China, Tian Yuan Foundation (11226067), the Fundamental Research Funds for the Central Universities (WM1214063), and China Postdoctoral Science Foundation (2012M511014).

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