Completing a 2 × 2 Block Matrix of Real Quaternions with a Partial Specified Inverse

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.


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 R. Suppose  1 ,  2 ,  1 ,  2 are nonnegative integers, and the solution and the expression for  22 were obtained in [1].Dai [2] considered this form of completion problems with symmetric and symmetric positive definite matrices over R.
Some other particular forms for 2 × 2 block matrices over R 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][5][6]).Quaternion matrices are receiving much attention as witnessed recently (e.g., [7][8][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: where H × denotes the set of all  ×  matrices over H and  −1 denotes the inverse matrix of .
Throughout, over the real quaternion algebra H, 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 H 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.R() 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 H, and the general expression for  11 is also obtained.In Section 3, we present a numerical example to illustrate the developed theory.

Main Results
In this section, we begin with the following lemmas.
Let H   denote the collection of column vectors with  components of quaternions and  be an  ×  quaternion matrix.Then the solutions of  = 0 form a subspace of H   of dimension ().We have the following lemma.
are inverse to each other, so we may suppose that ( 11 ) < ( 22 ).
If ( 22 ) = 0, necessarily ( 11 ) = 0 and we are finished.Let ( 22 ) =  > 0, then there exists a matrix  with  right linearly independent columns, such that  22  = 0.Then, using we have From we have It follows that the rank ( 12 ) ≥ .In view of (12), this implies Thus Lemma 3 (see [10]).Let  ∈ H × ,  ∈ H × ,  ∈ H × be known and  ∈ H × unknown.Then the matrix equation is consistent if and only if In that case, the general solution is where  1 ,  2 are any matrices with compatible dimensions over H.
By Lemma 1, let the singular value decomposition of the matrix  22 and  11 in Problem 1 be where Λ = diag( ).In that case, the general solution has the form of where  is an arbitrary matrix in H ( 2 −)× and  is an arbitrary matrix in H implying that (b) is satisfied.Conversely, from (c), we know that there exists a matrix  ∈ H  2 × 1 such that Let From ( 20), (21), and (26), we have It follows that This implies that Comparing corresponding blocks in (30), we obtain Let  *  = K.From ( 29), (30), we have In the same way, from (d), we can obtain We simplify the equation above.The left hand side reduces to ( +  12 ) 1  * 1 and so we have So, This implies that so that where  is an arbitrary matrix in H ( 2 −)× and  is an arbitrary matrix in H  1 × 1 .

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: