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Journal of Applied Mathematics
Volume 2013, Article ID 271978, 5 pages
http://dx.doi.org/10.1155/2013/271978
Research Article

Completing a Block Matrix of Real Quaternions with a Partial Specified Inverse

1Department of Mathematics, Shanghai University, Shanghai 200444, China
2School of Mathematics and Statistics, Suzhou University, Suzhou 234000, China

Received 4 December 2012; Revised 23 February 2013; Accepted 20 March 2013

Academic Editor: K. Sivakumar

Copyright © 2013 Yong Lin and Qing-Wen Wang. 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

This paper considers a completion problem of a nonsingular block matrix over the real quaternion algebra : Let be nonnegative integers, , and be given. We determine necessary and sufficient conditions so that there exists a variant block entry matrix such that is nonsingular, and is the upper left block of a partitioning of . The general expression for is also obtained. Finally, a numerical example is presented to verify the theoretical findings.

1. Introduction

The problem of completing a block-partitioned matrix of a specified type with some of its blocks given has been studied by many authors. Fiedler and Markham [1] considered the following completion problem over the real number field . Suppose are nonnegative integers, , and . Determine a matrix such that is nonsingular and is the lower right block of a partitioning of . This problem has the form of and the solution and the expression for were obtained in [1]. Dai [2] considered this form of completion problems with symmetric and symmetric positive definite matrices over .

Some other particular forms for block matrices over have also been examined (see, e.g., [3]), such as

The real quaternion matrices play a role in computer science, quantum physics, and so on (e.g., [46]). Quaternion matrices are receiving much attention as witnessed recently (e.g., [79]). Motivated by the work of [1, 10] and keeping such applications of quaternion matrices in view, in this paper we consider the following completion problem over the real quaternion algebra:

Problem 1. Suppose are nonnegative integers, , and . Find a matrix such that is nonsingular, and is the upper left block of a partitioning of . That is where denotes the set of all matrices over and denotes the inverse matrix of .

Throughout, over the real quaternion algebra , we denote the identity matrix with the appropriate size by , the transpose of by , the rank of by , the conjugate transpose of by , a reflexive inverse of a matrix over by which satisfies simultaneously and . Moreover, , where is an arbitrary but fixed reflexive inverse of . Clearly, and are idempotent, and each is a reflexive inverse of itself. denotes the right column space of the matrix .

The rest of this paper is organized as follows. In Section 2, we establish some necessary and sufficient conditions to solve Problem 1 over , and the general expression for is also obtained. In Section 3, we present a numerical example to illustrate the developed theory.

2. Main Results

In this section, we begin with the following lemmas.

Lemma 1 (singular-value decomposition [9]). Let be of rank . Then there exist unitary quaternion matrices and such that where and the 's are the positive singular values of .

Let denote the collection of column vectors with components of quaternions and be an quaternion matrix. Then the solutions of form a subspace of of dimension . We have the following lemma.

Lemma 2. Let be a partitioning of a nonsingular matrix , and let be the corresponding (i.e., transpose) partitioning of . Then .

Proof. It is readily seen that are inverse to each other, so we may suppose that .
If , necessarily and we are finished. Let , then there exists a matrix with right linearly independent columns, such that . Then, using we have
From we have It follows that the rank . In view of (12), this implies
Thus

Lemma 3 (see [10]). Let , , be known and unknown. Then the matrix equation is consistent if and only if In that case, the general solution is where , are any matrices with compatible dimensions over .

By Lemma 1, let the singular value decomposition of the matrix and in Problem 1 be where is a positive diagonal matrix, are the singular values of , , is a positive diagonal matrix, are the singular values of and .

, , , are unitary quaternion matrices, where , , ,  and  .

Theorem 4. Problem 1 has a solution if and only if the following conditions are satisfied:(a),(b),  that is ,(c),(d).
In that case, the general solution has the form of where is an arbitrary matrix in and is an arbitrary matrix in .

Proof. If there exists an matrix such that is nonsingular and is the corresponding block of , then is satisfied. From , we have that so that and are satisfied.
By (11), we have From Lemma 2, Notice that is the corresponding partitioning of , we have implying that is satisfied.
Conversely, from (c), we know that there exists a matrix such that Let From (20), (21), and (26), we have
It follows that
This implies that
Comparing corresponding blocks in (30), we obtain
Let . From (29), (30), we have
In the same way, from (d), we can obtain Notice that in (a) is a full column rank matrix. By (20), (21), and (33), we have so that It follows from (b) and (35) that is a full column rank matrix, so it is nonsingular.
From , we have the following matrix equation: that is where , were given, (from (27)). By Lemma 3, the matrix equation (37) has a solution if and only if By (21), (27), (32), and (33), we have that (38) is equivalent to: We simplify the equation above. The left hand side reduces to and so we have So, This implies that so that So, and hence, Finally, we obtain Multiplying both sides of (46) by from the left, considering (33) and the fact that is nonsingular, we have From Lemma 3, (38), (47), Problem 1 has a solution and the general solution is where is an arbitrary matrix in and is an arbitrary matrix in .

3. An Example

In this section, we give a numerical example to illustrate the theoretical results.

Example 5. Consider Problem 1 with the parameter matrices as follows:
It is easy to show that , are satisfied, and that so , are satisfied too. Therefore, we have where We also have
By Theorem 4, for an arbitrary matrices , we have it follows that The results verify the theoretical findings of Theorem 4.

Acknowledgments

The authors would like to give many thanks to the referees and Professor K. C. Sivakumar for their valuable suggestions and comments, which resulted in a great improvement of the paper. This research was supported by Grants from the Key Project of Scientific Research Innovation Foundation of Shanghai Municipal Education Commission (13ZZ080), the National Natural Science Foundation of China (11171205), the Natural Science Foundation of Shanghai (11ZR1412500), the Discipline Project at the corresponding level of Shanghai (A. 13-0101-12-005), and Shanghai Leading Academic Discipline Project (J50101).

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