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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 457298, 7 pages
On Extremal Ranks and Least Squares Solutions Subject to a Rank Restriction
Department of Mathematics and Computational Science, Huainan Normal University, Anhui 232038, China
Received 16 February 2014; Revised 11 June 2014; Accepted 22 June 2014; Published 8 July 2014
Academic Editor: Sofiya Ostrovska
Copyright © 2014 Hongxing Wang and Yeguo Sun. 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.
We discuss the feasible interval of the parameter and a general expression of matrix which satisfies the rank equation . With these results, we study two problems under the rank constraint . The first one is to determine the maximal and minimal ranks under the rank constraint . The second one is to derive the least squares solutions of under the rank constraint .
We adopt the following notation in this paper. The set of matrices with complex entries is denoted by . The conjugate transpose of a matrix is denoted by . The symbols and are the identity matrix and the rank of , respectively. stands for the matrix Frobenius norm. The Moore-Penrose inverse of is defined as the unique matrix satisfying and is denoted by (see ). Furthermore, we denote and .
In the literature, ranks of solutions of linear matrix equations have been studied widely. Uhlig  derived the extremal ranks of solutions of the consistent matrix equation of . Tian  derived the extremal ranks of solutions of . Li and Liu  studied the extremal ranks of Hermitian solutions of . Li et al.  studied the extremal ranks of solutions with special structure of . Liu  derived the extremal ranks of solutions of . Wang and Li  established the maximal and minimal ranks of the solution to consistent system , and . Wang and He  derived the extremal ranks of the general solution of the mixed Sylvester matrix equations Liu  derived the extremal ranks of least square solutions to . Sou and Rantzer  studied the minimum rank matrix approximation broblem in the spectral norm Wei and Shen  studied a more general problem where and . More results and applications about ranks of matrix expressions and solutions of matrix equations can be seen in ([2, 3, 8, 11–13], etc.).
Motivated by the work of [2, 3, 7–9, 14, 15], we consider a general problem. Assume that is a prescribed nonnegative integer and , , and are given matrices. We now investigate the problem to determine the maximal and minimal ranks of solutions to the rank equation . This problem can be stated as follows.
Problem 1. Given matrices , , and and nonnegative integer , characterize the set and determine the maximal and minimal ranks of solutions of the rank equation .
In [16–18], Wang, Wei, and Zha studied least squares solutions of line matrix equations under rank constraints, respectively. In , Wei and Wang derived a rank- Hermitian nonnegative definite least squares solution to the equation . In Problem 2, we discuss the least squares solutions of subject to . This problem can be stated as follows.
Problem 2. Given matrices , , and and nonnegative integer , determine the range of , such that there exists a least squares solution of subject to ; that is, characterize the set
The paper is organized as follows. In Section 2, we provide some preliminary results; in Sections 3 and 4, we study Problems 1 and 2, respectively; and finally in Section 5, we conclude the paper with some remarks.
Lemma 3 (see ). Let , , , and be given. Then where and .
Lemma 4. Let be given. Then
Lemma 5 (see ). Let , , , and be given. Then
Lemma 6 (see [22, 23] (the Eckart-Young-Mirsky theorem)). Let , be a given nonnegative integer in which and the singular value decomposition  of be where , and and are unitary matrices of appropriate sizes. Then Furthermore, when , when and , where is an arbitrary matrix satisfying and .
3. Solutions to Problem 1
Suppose that the matrices , , and are given. Let Then from  there exists such that , if and only if Furthermore, let be singular value decompositions of and with unitary matrices , , , and . Write in partitioned form as where , , , and . Also assume that the singular value decomposition of and the corresponding decompositions are given by where and are unitary matrices of appropriate sizes in which , , , , , , , , and .
We have the following result.
Theorem 7. Suppose that the singular value decompositions of matrices , , , , and are given in (19)–(21). , , , , and have the forms in (20) and (21). If satisfies (18), then any solution to the rank equation has the form where , , , , , and are arbitrary, , and .
Proof. From the singular value decompositions of matrices of , and , we observe that
Then by repeated application of Lemma 3, we have
in which , , , , , , , and . It follows that
Since , from (28c), we obtain
The identity follows by substituting (24)–(26) into (29). Hence, any solution to the rank equation has the form where .
Substituting (30) into the second partitioned matrix in (27), we obtain where . The expression of in (22) follows by substituting (31) into the first partitioned matrix in (27).
We have the following result.
Proof. From a general expression of for the rank equation given in (22), (10), and (34), we obtain
From (11), (22), and (34), we obtain Since and , we see that and . To simplify expression (38) by the two inequalities, we obtain expression (36) for the maximal rank of solutions to the rank equation .
Remark 9 (see ). Let , , and be as in Theorem 7. The matrix equation is consistent, if and only if there exists such that . Therefore, applying Theorem 8, we have the extremal ranks of solutions to the matrix equation :
Remark 10 (see ). Let , , and be as in Theorem 7 and let . Since , if and only if , and the matrix equation is always consistent, we can use , , and to replace , , and in (39). Then we have the extremal ranks of least squares solutions of the matrix equation :
In , Liu and Tian derive the extremal ranks of submatrices in a Hermitian solution to the consistent matrix equation . In the following theorem, we derive the range of such that there exists a Hermitian solution to the rank equation , and the maximal and minimal ranks of which may be proved in the same way as Theorem 8.
Theorem 11. Let and be given, and let be Hermitian. Then from  there exists a Hermitian matrix satisfying , if and only if If satisfies the above inequalities, then
4. Solutions to Problem 2
Let It is obvious that and there do not exist the least squares solutions of subject to . Therefore, we should study the range of , such that there exists a least squares solution of subject to .
Theorem 12. Let , , and be as in Theorem 7. Then there exists a least squares solution of under the rank constraint , if and only if
Proof. Let , , , , , , and be as in Theorem 7, and let be partitioned in the form
where , , , and . Let have the singular value decomposition
where , and and are unitary matrices of appropriate sizes.
From the partitioned form for in (21), Since the Frobenius norm is invariant, we have the following identities by substituting (22) into and applying (46) and (48): Therefore, there exists a least squares solution satisfying subject to if and only if , that is, if and only if
From the partitioned form for in (46) and the decompositions of and in (21), we have . Applying (9) gives The identity follows from applying the decompositions of and in (19) and the partitioned form for in (20). Substituting the decomposition of in (19) into , applying the partitioned forms for and in (21), we conclude that Hence, It follows from applying (12) that
Substituting (24)–(26) and (55) into (54), we have Therefor, the inequalities in (45) follow from substituting (17) and (56) into (50).
Theorem 13. Let , , , , , , , , , and be as in Theorem 7, and let and be partitioned as in (46) and (47), respectively. If satisfies (45), then any least squares solution satisfying has the form where , , , , , , and are arbitrary matrices, such that .(1)When , (2)when and , where is an arbitrary matrix satisfying and .
We first derived the expression of solutions to when Problem 1 is solvable. Based on these results, we obtained the extremal ranks of the expression of solutions to Problem 1, the solvability conditions of Problem 2, and the expression of least squares solutions when Problem 2 is solvable.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the referees for their helpful comments and suggestions. The work of the first author was supported in part by the National Natural Science Foundation of China (Grant no. 11171226). The work of the second author was supported in part by the University Natural Science Foundation of Anhui Province (Grant no. KJ2013A239) and the National Natural Science Foundation of China (Grant no. 11301529).
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