Journal of Applied Mathematics

Volume 2013 (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

Yong Lin1,2 and Qing-Wen Wang¹

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., [4–6]). Quaternion matrices are receiving much attention as witnessed recently (e.g., [7–9]). 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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