Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 8673809, 13 pages
http://dx.doi.org/10.1155/2016/8673809

Research Article

Explicit Determinantal Representation Formulas of -Weighted Drazin Inverse Solutions of Some Matrix Equations over the Quaternion Skew Field

Ivan I. Kyrchei

Pidstrygach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, Lviv 79060, Ukraine

Received 25 January 2016; Accepted 24 April 2016

Academic Editor: Romain Aubry

Copyright © 2016 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.

By using determinantal representations of the -weighted Drazin inverse previously obtained by the author within the framework of the theory of the column-row determinants, we get explicit formulas for determinantal representations of the -weighted Drazin inverse solutions (analogs of Cramer’s rule) of the quaternion matrix equations , , and .

1. Introduction

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

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

For , we denote by

  the column right space of ;  the right null space of ;  the column left space of ;  the left null space of .

In the past, researches into the quaternion skew field had a more theoretical importance, but now a growing number of investigations give wide practical applications of quaternions. In particular through their attitude orientation, the quaternions arise in various fields such as quaternionic quantum theory [1], fluid mechanics and particle dynamics [2, 3], computer graphics [4], aircraft orientation [5], robotic systems [6], and life science [7, 8].

Research on quaternion matrix equations and generalized inverses, which are usefulness tools used to solve matrix equations, has been actively ongoing for more recent years. We mention only some recent papers. Yuan et al. [9] derived solutions of the quaternion matrix equation and their applications in color image restoration. Wang et al. [10] studied extreme ranks of real matrices in solution of the quaternion matrix equation . Yuan et al. [11] obtained the expressions of least squares Hermitian solution with minimum norm of the quaternion matrix equation . Feng and Cheng [12] gave a clear description of the solution set to the quaternion matrix equation . Jiang and Wei [13] derived the explicit solution of the quaternion matrix equation . Caiqin et al. [14] obtained the expressions of the explicit solutions of quaternion matrix equations and . Yuan and Wang [15] gave the expressions of the least squares -Hermitian solution with the least norm of the quaternion matrix equation . Zhang et al. derived [16] the expressions of the minimal norm least squares solution, the pure imaginary least squares solution, and the real least squares solution for the quaternion matrix equation .

The definitions of the generalized inverse matrices have been extended to quaternion matrices as follows.

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

For with being the smallest positive number such that , the Drazin inverse of , denoted by , is defined to be the unique matrix that satisfies (3) and the equations

In particular, when , then is called the group inverse of and is denoted by . If , then is nonsingular, and .

Cline and Greville [17] extended the Drazin inverse of square matrix to rectangular matrix that has been generalized to the quaternion algebra as follows.

For and , the -weighted Drazin inverse of with respect to is the unique solution to the following equations:

where .

The Drazin inverse and weighted Drazin inverse have several important applications such as applications in singular differential and difference equations [18], signal processing [19], Markov chains and statistic problems [20, 21], descriptor continuous-time systems [22], numerical analysis and Kronecker product systems [23], solving singular fuzzy linear system [24, 25], constrained linear systems [26], and so forth.

Cramer’s rule for the -weighted Drazin inverse solutions, in particular, has been derived in [27] for singular linear equations and in [26] for a class of restricted matrix equations. Recently, within the framework of the theory of the column-row determinants, Song [28] has first obtained a determinantal representation of the -weighted Drazin inverse and Cramer’s rule of a class of restricted matrix equations over the quaternion algebra. But in obtaining, he has used auxiliary matrices other than that are given. In [29], we have obtained new determinantal representations of the -weighted Drazin inverse over the quaternion skew field without any auxiliary matrices.

An important application of determinantal representations of generalized inverses is the Cramer rule for generalized inverse solutions of matrix equations.

But when is there a need for a -weighted Drazin inverse solution? Consider, for example, the following matrix equation: . Let be rectangular and we can represent it as , where and are quadratic and singular. Furthermore, we have the following restrictions: . Then its -weighted Drazin inverse solution is needed.

In the paper we investigate analogs of Cramer’s rule for -weighted Drazin inverse solutions of the following quaternion matrix equations:

The paper is organized as follows. We start with introducing of the row-column determinants and determinantal representations of the Moore-Penrose and Drazin inverses for a quaternion matrix obtained by them in Section 2.1. Determinantal representations of the -weighted Drazin inverse and its properties were considered in Section 2.2. In Section 3.1, we give the background of the problem of Cramer’s rule for the -weighted Drazin inverse solution. In Section 3.2 we obtain explicit representation formulas of the -weighted Drazin inverse solutions (analogs of Cramer’s rule) of the quaternion matrix equation (10). Consequently, we get both similar and special determinantal representation formulas of the -weighted Drazin inverse solutions of (8) and (9). In Section 4, we give numerical examples to illustrate the main result.

2. Preliminaries

2.1. Determinantal Representations of the Moore-Penrose and Drazin Inverses by the Column and Row Determinants

The theory of the row-column determinants over the quaternion skew field has been introduced in [30–32], and later it has been applied to research generalized inverses and generalized inverse solutions of matrix equations. In particular, determinantal representations of the Moore-Penrose [33, 34] and explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations [35] and determinantal representations of the Drazin [36] and -weighted Drazin inverses [29] have been obtained by the author. Song derived determinantal representation of the generalized inverse [37], the Bott-Duffin inverse [38], the Cramer rule for the solutions of restricted matrix equations [39], the generalized Stein quaternion matrix equation [40], and so forth.

For we define row determinants and column determinants as follows.

Suppose that is the symmetric group on the set .

Definition 1. The th row determinant of is defined for all by putting

with conditions and for and .

Definition 2. The th column determinant of is defined for all by putting

with conditions and for and .

Suppose that denotes the submatrix of obtained by deleting both the th row and the th column. Let be the th column and let be the th row of . Suppose that denotes the matrix obtained from by replacing its th column with the column and that denotes the matrix obtained from by replacing its th row with the row .

The following theorem has a key value in the theory of the column and row determinants.

Theorem 3 (see [30]). If is Hermitian, then

Since all column and row determinants of a Hermitian matrix over are equal, we can define the determinant of a Hermitian matrix . By definition, we put , for all . The determinant of a Hermitian matrix has properties similar to a usual determinant. They are completely explored in [30, 31] by its row and column determinants. In particular, within the framework of the theory of the column-row determinants, we have the determinantal representation of the inverse matrix over by analogs of classical adjoint matrix. Further, we consider determinantal representations of generalized inverses obtained by the column-row determinants.

We will use the following notations. Let and be subsets of the order . By denote the submatrix of determined by the rows indexed by and the columns indexed by . Then denotes the principal submatrix determined by the rows and columns indexed by . If is Hermitian, then by denote the corresponding principal minor of . For , the collection of strictly increasing sequences of integers chosen from is denoted by . For fixed and , let .

Denote by and the th column and the th row of and by and the th column and the th row of , respectively.

The following theorem gives determinantal representations of the Moore-Penrose inverse over the quaternion skew field .

Theorem 4 (see [32]). If , then the Moore-Penrose inverse possesses the following determinantal representations:

or

for all and .

Proposition 5 (see [20]). If , then .

Denote by and the th column of and the th row of , respectively, for all . Using the determinantal representations of the Moore-Penrose inverse (13) and (14) and Proposition 5, the following determinantal representations of the Drazin inverse for an arbitrary square matrix over have been obtained in [33].

Theorem 6 (see [33]). If with and , then the Drazin inverse possesses the following determinantal representations:

In the special case, when is Hermitian, we can obtain simpler determinantal representations of the Drazin inverse.

Theorem 7 (see [33]). If is Hermitian with and , then the Drazin inverse possesses the following determinantal representations:

or

2.2. Determinantal Representations of the -Weighted Drazin Inverse

We introduce some mathematical background from the theory of the -weighted Drazin inverse [27, 41, 42] that can be generalized to .

Lemma 8. Let and with . Then one has

(a);(b); ;(c); ;(d),

where is the projector on along and is the projector on along .

In particular, the point (a) of Lemma 8 due to Cline and Greville [17] is generalized [28] to Using this proposition, we have obtained [29] the following determinantal representations of -weighted Drazin inverse.

Denote and .

Due to Theorem 6, we denote an entry of the Drazin inverse by

or

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

Then we have the following determinantal representations of :

where

for all and from (18) and from (19).

Similarly using ,

where the first factor is one of the following four possible equations:

for all , and an entry of the Drazin inverse is denoted by

or

where and are the th column of and the th row of , respectively, for all ; .

The point (c) of Lemma 8 due to [23] has been generalized to in [33]. Using this proposition, we have obtained the following two determinantal representations of the -weighted Drazin inverse.

Theorem 9 (see [29]). Let and with and . Then the -weighted Drazin inverse of with respect to possesses the following determinantal representations:

where , and , .

In the special cases, when and are Hermitian, we can obtain simpler determinantal representations of the -weighted Drazin inverse.

Theorem 10 (see [29]). If , , and are Hermitian with and , then the -weighted Drazin inverse with respect to possesses the following determinantal representations:

where is the th column of for all .

Theorem 11 (see [29]). If , , and are Hermitian with and , then the -weighted Drazin inverse with respect to possesses the following determinantal representations:

where is the th row of  for all .

3. Cramer’s Rule for the -Weighted Drazin Inverse Solution

3.1. Background of the Problem

In [27] Wei has established Cramer’s rule for solving of a general restricted equation:

where and with ,  and , . He proved if and , then (30) has a unique solution, , which can be presented by the following Cramer rule:

where and are matrices whose columns form bases for and , respectively.

Recently, within the framework of a theory of the column and row determinants, Song [28] has considered the characterization of the -weighted Drazin inverse over the quaternion skew and presented a Cramer rule of the restricted matrix equation:

where , , , , and with , , and , .

He proved that if

and there exist auxiliary matrices of full column rank, , , , and with additional terms of their ranges and null spaces, then the restricted matrix equation (32) has a unique solution:

Using auxiliary matrices, , , , and , Song presented its Cramer’s rule by analogy to (31).

In this paper we have avoided such approach and have obtained explicit formulas for determinantal representations of the -weighted Drazin inverse solutions of matrix equations by using only given matrices.

3.2. Cramer’s Rules for the -Weighted Drazin Inverse Solutions of Some Matrix Equations

Consider the matrix equation (32) with constraints (33). Denote and , where and .

Theorem 12. Suppose that , , and with , where , and , with , . If , , then the restricted matrix equation (32) has a unique solution:

which possesses the following determinantal representations for all and .
(i) Consider

where is the Drazin inverse of  and can be obtained by (23) and is the Drazin inverse of   and can be obtained by (21).
(ii) If and are Hermitian, then

or

where

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

Proof. The existence and uniqueness of solution (36) can be proved similarly as in [28], Theorem .
(i) To derive a Cramer rule (37) we use the point (a) from Lemma 8. Then we obtain

Denote , , and . Then (42) will be written componentwise as follows:

By changing the order of summation, from here it follows (37).
(ii) If and and and are Hermitian, then by Theorems 10 and 11 the -weighted Drazin inverses and possess the following determinantal representations, respectively:

where is the th column of for all and

where is the th row of for all .
By componentwise writing of (36) we obtain

Denote by the th column of for all . It follows from that

Suppose that and are, respectively, the unit row vector and the unit column vector whose components are , except the th components, which are . Substituting (47) and (45) into (46), we obtain

Since

then we have

Denote by

the th component of a row vector for all . Substituting it into (50), we have

Since , then it follows (39).
If we denote by

the th component of a column vector for all and substituting it into (50), we obtain

Since , then it follows (38).

Remark 13. To establish a Cramer rule of (32) we will not use the determinantal representations (28) and (28) for (36) because the corresponding determinantal representations of its solution will be too cumbersome. But they are suitable in the following corollaries.

Corollary 14. Suppose that the following restricted matrix equation is given:

where and with and . If and , then the restricted matrix equation (55)-(56) has a unique solution:

which possess the following determinantal representations for all and .
(i) Consider

where , is the th column of , , and .
(ii) Consider

where can be obtained by (23) and .
(iii) If is Hermitian, then

where is the th column of .

Proof. To derive a Cramer rule (58), we use the determinantal representation (27) for . Then

Denote , where . Since

where is the th column of , then (58) follows from (61).
Cramer’s rule (59) and (60) immediately follows from Theorem 12 by putting and .

Remark 15. In the complex case, that is, , , and , we substitute usual determinants for all corresponding row and column determinants in (58), (59), and (60).
Note that in case (iii), the condition being Hermitian is not necessary; then in the complex case (60) will have the form

where is the th column of .

Corollary 16. Suppose that the following restricted matrix equation is given:

where and with and . If and , then the restricted matrix equation (64) has a unique solution:

which possesses the following determinantal representations for and .
(i) Consider

where , is the th row of , is the th row of , and .
(ii) Consider

where can be obtained by (21) and .
(iii) If is Hermitian, then

where is the th row of for all .

Proof. The proof is similar to the proof of Corollary 14 in the point (i) and follows from Theorem 12 by putting and .

Remark 17. In the complex case, that is, , , and , we substitute usual determinants for all corresponding row and column determinants in (66), (67), and (68). Herein the condition being Hermitian is not necessary; then in the complex case (68) can be represented as follows:

where is the th row of for all .

4. Examples

In this section, we give examples to illustrate our results.

(1) Let us consider the matrix equation

with the restricted conditions (56), where

Then

and , , , and . So, , , and .

We will find the -weighted Drazin inverse solution of (70) by its determinantal representation (58). We have

Since by (58)

where

then

We finally get

(2) Let now us consider the matrix equation

with constraints (33), where

Since the following matrices are Hermitian:

then we can find the -weighted Drazin inverse solution of (78) by its determinantal representation (38).

We have

and and . Since

then

We have

By (40), we can get

Since

then finally we obtain

Similarly,

So, the -weighted Drazin inverse solution of (78) is

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

Competing Interests

The author declares that there are no competing interests.

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