Journal of Applied Mathematics

Volume 2013, Article ID 271978, 5 pages

http://dx.doi.org/10.1155/2013/271978

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

^{1}Department of Mathematics, Shanghai University, Shanghai 200444, China^{2}School 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).

#### References

- M. Fiedler and T. L. Markham, “Completing a matrix when certain entries of its inverse are specified,”
*Linear Algebra and Its Applications*, vol. 74, pp. 225–237, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Dai, “Completing a symmetric $2\times 2$ block matrix and its inverse,”
*Linear Algebra and Its Applications*, vol. 235, pp. 235–245, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. W. Barrett, M. E. Lundquist, C. R. Johnson, and H. J. Woerdeman, “Completing a block diagonal matrix with a partially prescribed inverse,”
*Linear Algebra and Its Applications*, vol. 223/224, pp. 73–87, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. L. Adler,
*Quaternionic Quantum Mechanics and Quantum Fields*, Oxford University Press, New York, NY, USA, 1994. View at Zentralblatt MATH · View at MathSciNet - A. Razon and L. P. Horwitz, “Uniqueness of the scalar product in the tensor product of quaternion Hilbert modules,”
*Journal of Mathematical Physics*, vol. 33, no. 9, pp. 3098–3104, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. W. Shuai, “The quaternion NN model: the identification of colour images,”
*Acta Comput. Sinica*, vol. 18, no. 5, pp. 372–379, 1995 (Chinese). View at Google Scholar - F. Zhang, “On numerical range of normal matrices of quaternions,”
*Journal of Mathematical and Physical Sciences*, vol. 29, no. 6, pp. 235–251, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Z. Zhang,
*Permanent Inequalities and Quaternion Matrices [Ph.D. thesis]*, University of California, Santa Barbara, Calif, USA, 1993. - F. Zhang, “Quaternions and matrices of quaternions,”
*Linear Algebra and Its Applications*, vol. 251, pp. 21–57, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q.-W. Wang, “The general solution to a system of real quaternion matrix equations,”
*Computers & Mathematics with Applications*, vol. 49, no. 5-6, pp. 665–675, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet