Journal of Mathematics
Volume 2018 (2018), Article ID 6294672, 12 pages
https://doi.org/10.1155/2018/6294672

Research Article

Determinantal Representations of Solutions and Hermitian Solutions to Some System of Two-Sided Quaternion Matrix Equations

Ivan I. Kyrchei

Pidstrygach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, Lviv, Ukraine

Correspondence should be addressed to Ivan I. Kyrchei; st260664@gmail.com

Received 5 April 2018; Accepted 27 September 2018; Published 1 November 2018

Academic Editor: Frank Uhlig

Copyright © 2018 Ivan I. Kyrchei. 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.

Within the framework of the theory of quaternion row-column determinants previously introduced by the author, we derive determinantal representations (analogs of Cramer’s rule) of solutions and Hermitian solutions to the system of two-sided quaternion matrix equations and . Since the Moore-Penrose inverse is a necessary tool to solve matrix equations, we use determinantal representations of the Moore-Penrose inverse previously obtained by the theory of row-column determinants.

1. Introduction

The study of matrix equations and systems of matrix equations is an active research topic in matrix theory and its applications. Research on the classical system of two-sided matrix equations

over the complex field, a principle domain, and the quaternion skew field has been actively ongoing for many years. For instance, Mitra [1, 2] gave necessary and sufficient conditions of system (3) over the complex field and the expression for its general solution. Navarra et al. [3] derived a new necessary and sufficient condition for the existence and a new representation of the general solution to (3) over the complex field and used the results to give a simple representation. Özgüler et al. [4] gave solutions to (3) over a principle domain. Wang [5] got solvability conditions of system (3) over the quaternion skew field and represented its general solution in terms of generalized inverses.

Since quaternion matrices play an important role in quantum mechanics, signal processing, and control theory, research on quaternion matrix equations and systems of quaternion matrix equations, their general solutions, especially Hermitian solutions, has been actively developing for more recent years (see, e.g., [6–26]).

Throughout the paper, we denote the real number field by , the set of all matrices over the quaternion algebra

by , and by its subset of matrices of a rank . Let be the ring of quaternion matrices. For , the symbol stands for the conjugate transpose (Hermitian adjoint) matrix of . The matrix is Hermitian if .

Motivated by the wide application of quaternion matrix equations and in order to improve the theoretical development of solutions and Hermitian solutions to quaternion matrix equations, we consider a special case of (3), more specifically,

Generalized inverses are useful tools used to solve matrix equations. The definition of the Moore-Penrose inverse matrix has been extended to quaternion matrices as follows.

Definition 1. The Moore-Penrose inverse of , denoted by , is the unique matrix satisfying the following four equations:

The main goal of this paper is to derive determinantal representations of solutions and Hermitian solutions to system (1) over the quaternion skew field using previously obtained determinantal representations of the Moore-Penrose inverse. Evidently, determinantal representation of a solution gives a direct method of its finding analogous to the classical Cramer’s rule that has important theoretical and practical significance [27].

Through the noncommutativity of the quaternion algebra when difficulties arise already in determining the quaternion determinant, the problem of the determinantal representation of generalized inverses only now can be solved due to the theory of column-row determinants introduced in [28, 29]. Within the framework of the theory of column-row determinants, determinantal representations of various kinds of generalized inverses (generalized inverses) solutions of quaternion matrix equations have been derived by the author (see, e.g., [30–39]) and by other researchers (see, e.g., [40–43]).

The paper is organized as follows. In Section 2, we start with preliminary introduction of row-column determinants, determinantal representations of the Moore–Penrose inverse previously obtained within the framework of the theory of row-column determinants, and Cramer’s rules for the two-sided matrix equation and of its special cases, left- and right-sided equations. We derive some simplified expressions of general and partial solutions to (3), a solvability criterion and expressions of general and partial solutions to system (1), and determinantal representations (analogs of Cramer’s rule) of its solution and Hermitian solution. A numerical example to illustrate the main results is considered in Section 4. Finally, the conclusion is drawn in Section 5.

2. Preliminaries

For , we define row determinants and column determinants. Suppose is the symmetric group on the set .

Definition 2 (see [28]). The th row determinant of is defined for all by putting

where and for all and .

Definition 3 (see [28]). The th column determinant of is defined for all by putting

where and for and .

Since [28] for Hermitian we have

the determinant of a Hermitian matrix is defined by putting for all .

The properties of row and column determinants are completely explored in [29]. We note the following that will be required below.

Lemma 4. Let . Then

We shall use the following notations. Let and be subsets of the order . Let be a submatrix of whose rows are indexed by and columns indexed by . Similarly, let be a principal submatrix of whose rows and columns are indexed by . If is Hermitian, then is the corresponding principal minor of . For , the collection of strictly increasing sequences of integers chosen from is denoted by . For fixed and , let .

Let be the th column and be the th row of . Suppose denotes the matrix obtained from by replacing its th column with column , and denotes the matrix obtained from by replacing its th row with the row . Denote by and the th column and the th row of , respectively.

Theorem 5 (see [30]). If , then the Moore-Penrose inverse has the following determinantal representations,

and

Remark 6. For an arbitrary full-rank matrix , we put

where a column vector and a row vector have appropriate sizes.

Remark 7. First note that . Because of symbol equivalence, we shall use the denotation as well. So, by Lemma 4, for the Hermitian adjoint matrix determinantal representations of its Moore-Penrose inverse are

and

Corollary 8. If , then the projection matrix has the determinantal representation

where is the th column of .

Corollary 9. If , then the projection matrix has the determinantal representation

where is the th row of .

The orthogonal projectors and induced by will be used below.

Theorem 10 (see [44]). Let , , be known and be unknown. Then the matrix equation

is consistent if and only if . In this case, its general solution can be expressed as

where , are arbitrary matrices over with appropriate dimensions.

Theorem 11 (see [31]). Let , . Then the partial solution to (16) has determinantal representations,

or

where

are the column vector and the row vector, respectively. and are the th row and the th column of .

Corollary 12. Let , be known and be unknown. Then the matrix equation is consistent if and only if . In this case, its general solution can be expressed as , where is an arbitrary matrix over with appropriate dimensions. The partial solution has the following determinantal representation,

where is the th column of .

Corollary 13. Let , be given and be unknown. Then the equation is solvable if and only if and its general solution is , where is any matrix with conformable dimension. Moreover, its partial solution has the determinantal representation,

where is the th row of .

3. Cramer’s Rules for the Solution and Hermitian Solution to System (3)

First, consider the general system (1).

Lemma 14 (see [5]). Let , , , , , be given and is to be determined. Put , , , . Then system (1) is consistent if and only if

In that case, the general solution to (1) can be expressed as the following:

where and are arbitrary matrices over with compatible dimensions.

Some simplification of (25) can be derived due to the quaternionic analogue of the following proposition.

Lemma 15 (see [45]). If is Hermitian and idempotent, then the following equation holds for any matrix ,

It is evident that if is Hermitian and idempotent, then the following equation is true as well:

Since , , and are projectors, then by (26) and (27), we have, respectively,

Using (28) and (23), we obtain the following expressions of (25):

By putting , as zero-matrices in (29), we obtain the following partial solution of (25):

Now consider system (1). Since

so , , and for . Moreover, substituting , we have ; similarly, . Due to the above, we obtain the following analog of Lemma 14.

Lemma 16. Let , , , be given and is to be determined. Then system (3) is consistent if and only if

In that case, the general solution to (3) can be expressed as follows:

and are arbitrary matrices over with compatible dimensions.

By putting , as zero-matrices, the partial solution of (3) is

Further we give determinantal representations of (34).

Let , , , , , and . There exist , , , , , and .

Consider each term of (34) separately.

(i) Denote . For the first term of (34) , we have

By using determinantal representations (9) and (13) of the Moore-Penrose inverses and , respectively, we obtain

Suppose and are the unit row vector and the unit column vector, respectively, such that all their components are , except the th components, which are . Since , then

If we denote by

the th component of a row vector , then

Further, it is evident that , so the first term of (34) has the determinantal representation

where

If we denote by

the th component of a column vector , then

So, another determinantal representation of the first term of (34) is

where

are the column vector.

(ii) Similarly above, for the second term of (34), we have

or

where

are the column vector and the row vector, respectively. and are the th row and the th column of . Note that

(iii) The third term of (34) can be obtained similarly as well. So,

or

where

are the column vector and the row vector, respectively. are the th row, and are the th column of . The following expressions give some simplifications in computing. Since and , then .

(iv) Using (9) for determinantal representations of and in the fourth term of (34), we obtain

where , are the th columns of the matrices and , respectively; is the first term; is the th element of with determinantal representation by (14) as

where is the th column of . Note that and .

(v) Similarly to the previous case,

(vi) Consider the sixth term by analogy to the fourth term. So,

where

and

are the column vector and the row vector, respectively. and are the th row and the th column of .

(vii) Using (9) for determinantal representations of and (12) for in the seventh term of (34), we obtain

where , are the th column of and the th row of , respectively.

Hence, we prove the following theorem.

Theorem 17. Let , , and , . Then for the partial solution (34) to system (3), we have

where the term has the determinantal representations (40) and (44); , (46) and (47); , (49) and (50); , (52); , (54); , (55); and , (58).

Due to Khatri and Mitra [46], the next lemma can be generalized to .

Lemma 18. Let and and be known and be unknown. Then the matrix equation

has a Hermitian solution if and only if . In that case, the general Hermitian solution of (60) is

where is any matrix.

As it follows from the above, if , then the determinantal representation of the partial Hermitian solution is

where

are the row vector and the column vector and and are the th column and the th row of , respectively.

The general Hermitian solution to system (3) can be expressed as , where is an arbitrary solution of (3). Since by Lemma 18, the existence of Hermitian solutions (1) needs and to be Hermitian, then

So, the determinantal representation of the partial Hermitian solution can be obtained as for all , where is determined by Theorem 17 and so that

where

are the row vector and the column vector, respectively. and are the th column and the th row of ;

where

are the row vector and the column vector, respectively. are the th column and are the th row of ;

where is the row of and is the th row of , respectively; can be obtained by (40) or (44); evidently, and is the th element of with determinantal representation by (14) as

where is the th column of ;

where

and

are the row vector and the column vector, respectively. and are the th column and the th row of ;

where , are the th row of and the th column of , respectively.

4. An Example

In this section, we give an example to illustrate our results. Let us consider the system of matrix equations

where

Since

then , . By Theorem 5, one can find

Since , , , then, by Lemma 16, system (75) is consistent.

First, we can find the solution of (75) by direct calculation. Since , where

then

Now, we find the solution of (75) by its determinantal representation by Theorem 17. Since

then by (41)

Similarly, , . So, . Further, by (40),

Since

and , then

So, by (47) and by (49), and , respectively. Moreover, for all . So, .

So, obtained by Cramer’s rule and the matrix method (80) are equal.

Similarly, we can obtain for all the remainder solutions.

Note that we used Maple with the package CLIFFORD in the calculations.

5. Conclusions

Within the framework of the theory of quaternion row-column determinants previously introduced by the author, we have derived determinantal representations (analogs of Cramer’s rule) of the general and Hermitian solutions to the system of two-sided quaternion matrix equations and . Since the Hermitian solution is , where is an arbitrary solution, the determinantal representation of is derived as well. To accomplish that goal, we have used the determinantal representations of the Moore-Penrose matrix inverse which were previously introduced by the author.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

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