Abstract and Applied Analysis

Volume 2014 (2014), Article ID 457298, 7 pages

http://dx.doi.org/10.1155/2014/457298

Research Article

On Extremal Ranks and Least Squares Solutions Subject to a Rank Restriction

Hongxing Wang and Yeguo Sun

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.

Abstract.

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 .

1. Introduction

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 [1]). Furthermore, we denote and .

In the literature, ranks of solutions of linear matrix equations have been studied widely. Uhlig [2] derived the extremal ranks of solutions of the consistent matrix equation of . Tian [3] derived the extremal ranks of solutions of . Li and Liu [4] studied the extremal ranks of Hermitian solutions of . Li et al. [5] studied the extremal ranks of solutions with special structure of . Liu [6] derived the extremal ranks of solutions of . Wang and Li [7] established the maximal and minimal ranks of the solution to consistent system , and . Wang and He [8] derived the extremal ranks of the general solution of the mixed Sylvester matrix equations

Liu [9] derived the extremal ranks of least square solutions to . Sou and Rantzer [10] studied the minimum rank matrix approximation broblem in the spectral norm

Wei and Shen [11] 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 [19], 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.

2. Preliminaries

In this section we present some preliminary results which will be used in the following sections to study Problems 1 and 2.

Lemma 3 (see [20]). Let , , , and be given. Then

where and .

Lemma 4. Let be given. Then

Lemma 5 (see [21]). 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 [24] 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

In this section, we study Problem 1 proposed in Section 1.

Suppose that the matrices , , and are given. Let

Then from [25] 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

Furthermore, write

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).

Let . From (28b), we have

Substituting (24) into the above identity, we have

By applying Lemma 3 (9) to the final identity in (23), it follows that

We have the following result.

Theorem 8. Let , , and be as in Problem 1 and let satisfy (18). Then

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 [3]). 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 [9]). 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 [14], 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 [15] there exists a Hermitian matrix satisfying , if and only if

If satisfies the above inequalities, then

4. Solutions to Problem 2

In this section, we study Problem 2 proposed in Section 1.

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 .

Proof. When satisfies the inequalities in (45), then, by applying Lemma 6 and (48), we obtain the desired form of in (58) and (59), respectively.

5. Conclusions

In this paper, we have discussed the solutions to Problem 1

and the solutions to Problem 2

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.

Acknowledgments

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).

References

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Springer, Berlin, Germany, 2nd edition, 2003.

F. Uhlig, “On the matrix equation with applications to the generators of a controllability matrix,” Linear Algebra and its Applications, vol. 85, pp. 203–209, 1987.

Y. Tian, “Ranks of solutions of the matrix equation ,” Linear and Multilinear Algebra, vol. 51, no. 2, pp. 111–125, 2003.

R. Li and Y. Liu, “Ranks of Hermitian solutions of the matrix equation ,” Far East Journal of Mathematical Sciences, vol. 26, no. 1, pp. 117–126, 2007.

Y. Li, F. Zhang, W. Guo, and J. Zhao, “Solutions with special structure to the linear matrix equation ,” Computers & Mathematics with Applications, vol. 61, no. 2, pp. 374–383, 2011.

Y. H. Liu, “Ranks of solutions of the linear matrix equation ,” Computers & Mathematics with Applications, vol. 52, no. 6-7, pp. 861–872, 2006.

Q.-W. Wang and C.-K. Li, “Ranks and the least-norm of the general solution to a system of quaternion matrix equations,” Linear Algebra and Its Applications, vol. 430, no. 5-6, pp. 1626–1640, 2009.

Q. Wang and Z. H. He, “Solvability conditions and general solution for mixed Sylvester equations,” Automatica, vol. 49, no. 9, pp. 2713–2719, 2013.

Y. H. Liu, “Ranks of least squares solutions of the matrix equation ,” Computers & Mathematics with Applications, vol. 55, no. 6, pp. 1270–1278, 2008.

K. C. Sou and A. Rantzer, “On a generalized matrix approximation problem in the spectral norm,” Linear Algebra and Its Applications, vol. 436, no. 7, pp. 2331–2341, 2012.

M. Wei and D. Shen, “Minimum rank solutions to the matrix approximation problems in the spectral norm,” SIAM Journal on Matrix Analysis and Applications, vol. 33, no. 3, pp. 940–957, 2012.

X. F. Duan, Q. W. Wang, and J. F. Li, “On the low-rank approximation arising in the generalized Karhunen-Loeve transform,” Abstract and Applied Analysis, vol. 2013, Article ID 528281, 8 pages, 2013.

H. Wang and J. Xu, “Some results on characterizations of matrix partial orderings,” Journal of Applied Mathematics, vol. 2014, Article ID 408457, 6 pages, 2014.

Y. Liu and Y. Tian, “Extremal ranks of submatrices in an Hermitian solution to the matrix equation with applications,” Journal of Applied Mathematics and Computing, vol. 32, no. 2, pp. 289–301, 2010.

Y. Tian and Y. Liu, “Extremal ranks of some symmetric matrix expressions with applications,” SIAM Journal on Matrix Analysis and Applications, vol. 28, no. 3, pp. 890–905, 2006.

H. Wang, “On least squares solutions subject to a rank restriction,” Linear and Multilinear Algebra, 2014.

M. Wei, “Perturbation theory for the Eckart-Young-Mirsky theorem and the constrained total least squares problem,” Linear Algebra and Its Applications, vol. 280, no. 1–3, pp. 267–287, 1998.

H. Y. Zha, “The restricted singular value decomposition of matrix triplets,” SIAM Journal on Matrix Analysis and Applications, vol. 12, no. 1, pp. 172–194, 1991.

M. Wei and Q. Wang, “On rank-constrained Hermitian nonnegative-definite least squares solutions to the matrix equation ,” International Journal of Computer Mathematics, vol. 84, no. 6, pp. 945–952, 2007.

G. Marsaglia and G. P. H. Styan, “Equalities and inequalities for ranks of matrices,” Linear and Multilinear Algebra, vol. 2, pp. 269–292, 1974.

Y. Tian, “More on maximal and minimal ranks of Schur complements with applications,” Applied Mathematics and Computation, vol. 152, no. 3, pp. 675–692, 2004.

C. Eckart and G. Young, “The approximation of one matrix by another of lower rank,” Psychometrika, vol. 1, no. 3, pp. 211–218, 1936.

L. Mirsky, “Symmetric gauge functions and unitarily invariant norms,” The Quarterly Journal of Mathematics, vol. 11, pp. 50–59, 1960.

G. H. Golub and C. F. van Loan, Matrix computations, vol. 3 of Johns Hopkins Series in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, Md, USA, 2nd edition, 1989.

D. L. Chu, H. C. Chan, and D. W. C. Ho, “Regularization of singular systems by derivative and proportional output feedback,” SIAM Journal on Matrix Analysis and Applications, vol. 19, no. 1, pp. 21–38, 1998.