The General Solution of Quaternion Matrix Equation Having <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5268pt" id="M1" height="12.3952pt" version="1.1" viewBox="-0.0657574 -7.8684 8.64031 12.3952" width="8.64031pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-229" d="M471 401C471 442 455 448 440 448C421 448 389 439 366 424C314 390 233 331 165 242H163L187 345C193 372 197 393 197 409C197 435 189 448 174 448C146 448 80 408 23 349L40 327C64 351 97 373 107 373C115 373 118 366 111 337L29 -4L36 -12C56 -4 83 3 110 9C123 69 136 126 150 172C227 283 336 381 377 381C392 381 396 370 379 298L326 74C290 -77 273 -172 273 -197C273 -219 316 -261 346 -261L353 -246C348 -227 352 -174 374 -66L457 310C467 352 471 381 471 401Z"/></g></svg>-</span>Skew-Hermicity and Its Cramer’s Rule

Rehman, Abdur; Kyrchei, Ivan; Ali, Ilyas; Akram, Muhammad; Shakoor, Abdul

doi:https://doi.org/10.1155/2019/7939238

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2019 | Article ID 7939238 | https://doi.org/10.1155/2019/7939238

The General Solution of Quaternion Matrix Equation Having -Skew-Hermicity and Its Cramer’s Rule

Abdur Rehman,¹Ivan Kyrchei,²Ilyas Ali,¹Muhammad Akram,¹and Abdul Shakoor³

Academic Editor: Raffaele Solimene

Received18 Mar 2019

Accepted10 Jun 2019

Published30 Jul 2019

Abstract

We determine some necessary and sufficient conditions for the existence of the -skew-Hermitian solution to the following system over the quaternion skew field and provide an explicit expression of its general solution. Within the framework of the theory of quaternion row-column noncommutative determinants, we derive its explicit determinantal representation formulas that are an analog of Cramer’s rule. A numerical example is also provided to establish the main result.

1. Introduction

Throughout this paper, the real number field and the complex field are denoted by and , respectively, and for the quaternion algebraThe set of all matrices of dimension over is represented by . An identity matrix with conformable size is denoted by . For any matrix over , and stand for the column right space and the row left space of , respectively. denotes the dimension of . By [1], we have , which is known as rank of and denoted by . For any quaternion , its conjugate is . So, represents the conjugate transpose of . means the Moore-Penrose inverse of , i.e., the exclusive matrix satisfying More results on generalized inverses can be seen in [2, 3]. Furthermore, and are couple of the projectors induced by , respectively. It is evident that and .

The notion of quaternions was first explored by an Irish mathematician Sir William Rowan Hamilton in [4]. Quaternions have ample use in diverse areas of mathematics like computation, geometry and algebra; see, e.g. [5–8]. The researcher in [9] used quaternion in the field of computer graphics. Very recently, quaternion matrices have secured a vital role in control theory, mechanics, altitude control, quantum physics and signal processing; see, e.g., [10, 11]. In skeletal animation systems, quaternions are mostly practiced to interpolate between joint orientations specified with key frames or animation curves [12]. The comprehensive study on quaternions can be found in [13].

The Hermitian solutions to some classical matrix equations were examined by numerous researchers in various papers by adopting different approaches. For instance, Khatri and Mitra [14] and Groß in [15] computed some necessary and sufficient condition for the Hermitian solution to some significant matrix equation with their general solution when they are consistent.

A matrix is said to be -Hermitian and skew--Hermitian [16, 17] if and , where , respectively. Applications of -Hermitian matrices in the fields of convergence analysis in statistical signal processing and linear modelling can be viewed in [16, 18, 19]. The singular value decomposition of the -Hermitian matrix was examined in [17]. Very recently, a researcher in [20] determined the skew--Hermitian solution to some significant matrix equations including and gave general solution to these equation when they are consistent.

He and Wang [21] gave the general solution of bearing -Hermicity over . An iterative algorithm for determining the -Hermitian and -skew-Hermitian solutions to the quaternion generalized Sylvester equation were established in [22]. For more related papers on -Hermicity, one may refer to [23–28]. Notice that Sylvester-type matrix equations have a huge amount of practical applications in feedback control [29, 30], robust control [31], pole/eigenstructure assignment design [32], neural network [33], and so on.

Keeping in view the latest development and the application of skew--Hermitian matrices in the fields like convergence analysis in statistical signal processing and linear modelling and quaternion matrices in mind, in this paper, we explore the -skew-Hermitian solution of the equation when it is consistent. We not only obtain the explicit form of the -skew-Hermitian solution in terms of generalized inverses but also give its explicit determinantal representation formulas that are an analog of Cramer’s rule. Our proposed Cramer’s rule is based on the theory of row-column noncommutative determinants introduced in [34], by using determinantal representations of the Moore-Penrose inverse matrix [35]. Within the framework of the theory of noncommutative row-column determinants, determinantal representations of various generalized quaternion inverses and generalized inverse solutions to quaternion matrix equations have been derived by one of the authors (see, e.g., [36–46]) and by other researchers (see, e.g., [47–51]).

Observe that the system (3) is a particular case of our system (5). It is noteworthy to say that the researcher in [20] made nice effort to calculate the -skew-Hermitian solution of (3) with its general solution but he did not represent the necessary and sufficient conditions in terms of rank equalities and we will overcome this issue in this paper as well. Moreover, we give explicit determinantal representation formulas for the solution to (5).

The remaining part of this paper is composed as follows. In Section 2, we start with some provisions from generalized inverses and its determinantal representations which have vital role during the construction of the main results of this paper. Necessary and sufficient conditions for the general solution to (5), where and are -skew-Hermitian, are presented in Section 3. A particular case of (5) is also examined in this section. In Section 4, we give Cramer’s rule to Eq.(5). Based on obtained explicit determinantal representation formulas, in Section 5, we present a numerical example to glorify the notion established in this paper. Finally, in Section 6, we provide a conclusion to this article.

2. Preliminaries. Some Provisions from Generalized Inverses and Its Determinantal Representations

We begin with some famous results which will be used in the remaining part of this paper.

Lemma 1 (see [52]). Let and be known. Then (1).(2).(3).

Lemma 2 (see [21]). Let be given. Then (1)(2)(3)(4)(5)(6)

Lemma 3 (see [53]). Let , and be given matrices with right sizes over . Then (1)(2)(3)

Remark 4. Since for any for all , and , then elements of the main diagonal of an η₁-Hermitian matrix must be as and a pair of elements which are symmetric with respect to the main diagonal can be represented as Similarly, elements of the main diagonal of an -skew-Hermitian matrix must be as and a pair of elements which are symmetric with respect to the main diagonal can be represented as where for all .

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

Definition 5 (see [34]). The th row determinant of is defined for all by putting where is the left-ordered permutation. It means that its first cycle from the left starts with , while other cycles start from the left with the minimal of all the integers which are contained in it, and the order of disjoint cycles (except for the first one) is strictly conditioned by increase from left to right of their first elements, .

Definition 6 (see [34]). The th column determinant of is defined for all by putting where is the right-ordered permutation. It means that its first cycle from the right starts with , while other cycles start from the right with the minimal of all the integers which are contained in it, and the order of disjoint cycles (except for the first one) is strictly conditioned by increase from right to left of their first elements, .

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

We state some properties of row-column determinants needed below.

Lemma 7 (see [34]). If the th row of is a left linear combination of other row vectors, i.e., , where and for all and , then

Lemma 8 (see [34]). If the th column of is a right linear combination of other column vectors, i.e., , where and for all and , then

Lemma 9 (see [34]). Let . Then , for all .

Since by Definitions 5 and 6 for for all , then, due to Lemma 9, the next lemma follows immediately.

Lemma 10. Let . Then for all

So, Lemma 10 gives the following features of -Hermitian and -skew-Hermitian matrices.

Lemma 11. If is -Hermitian, then If is -skew-Hermitian, then for all .

Remark 12. Since [34] for Hermitian we have the determinant of a Hermitian matrix is defined by putting for all .

Properties of the determinant of a Hermitian matrix are similar to the properties of an usual (commutative) determinant and they have been completely explored by row-column determinants in [34].

The following notation will be used for determinantal representations of the Moore-Penrose inverse. Let and be subsets with . Let be a submatrix of whose rows and columns are indexed by and , respectively. Similarly, let be a principal submatrix of whose rows and columns are indexed by . If is Hermitian, then denotes the corresponding principal minor of . The collection of strictly increasing sequences of integers chosen from is denoted by for . For fixed and , let , .

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

Theorem 13 (see [35]). If , then the Moore-Penrose inverse have the following determinantal representations or

Remark 14. Note that for an arbitrary full-rank matrix , a row-vector , and a column-vector we put, respectively, (i)if , then for all ;(ii)if , then for all .

Corollary 15. If , then the Moore-Penrose inverse , , and have the following determinantal representations, respectively,

Corollary 16 (see [35]). If , then the projection matrix has the determinantal representation where and are the th column and the th row of .

Corollary 17 (see [35]). If , then the projection matrix has the determinantal representation where and are the th row and the th column of .

Lemma 18 (see [54]). 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.

Lemma 19 (see [36]). Let , . Then the partial solution to (35) have determinantal representations, where are the column vector and the row vector, respectively. and are the th row and the th column of .

Corollary 20 (see [36]). Let , be known and be unknown. Then 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 21 (see [36]). Let , be given, and be unknown. Then is solvable if and only if and its general solution is , where is a any matrix with conformable dimension. Moreover, its partial solution has the determinantal representation, where is the th row of .

3. Some Solvability Conditions and the General Solution to (5)

In this section, we provide some necessary and sufficient conditions for the system (5) to have a solution , where and Furthermore, its general solution is also constructed when the solvability conditions are accomplished.

Theorem 22. Let and be given coefficient matrices in (5) over with conformable sizes. Denote Then (1)The system (5) has a solution , where and .(2)The coefficient matrices in (5) satisfy (3)(4)are equivalent statements. Under these conditions, the general solution to the system (5) can be demonstrated as where and are arbitrary matrices over .

Proof. Obviously, (2)(3).
Now we show (2) (4). By means of Lemmas 1 and 2, we have Similarly, Now we show (1): If the system (5) has a solution , where and . Then (2)(1): Now we prove that mentioned in (44)–(46), respectively, is a solution of (5) under (42). Obviously, and represented in (44) and (45) are -skew-Hermitian. Now putting and in the following expression and simplifying, we have Now we show that in (50). By Lemma 3, equation (42) and , we have By using (51), can be written as follows: i.e., By using (53) and , we can write By using , (42) and (51), the term in (54) can be expressed as By putting equation (55) in equation (54), we have Hence, the term in the parenthesis in (50) becomes zero and we have Consequently, mentioned in (44)–(46) is a solution of (5).
Next, we have to show that any solution , where and are -skew-Hermitian matrices, of the system (5) can be expressed by (44)–(46). Assume that , where and , be an arbitrary solution of For this ambition, we show that an arbitrary solution of (5) can be written like (44)–(46).
Note that From (59), we acquire that Put With the help of (62)-(63), can be written as follows: To show , we examine the term present in “+” of (64). Since and , we have The last term of (65) can be computed by using (60) as follows: The 2nd term in the last line of (66) can be written with the help of (51) as follows: With the help of (65)–(67) and (64), we get
By using , (59) and (63) in (53), we obtainFollowing the same pattern, we have Hence the general solution of the equation (5) can be expressed by (44)–(46). Thus the proof is done.

Now we discuss a particular case of (5). If is zero in (5) then we obtain the following renowned result.

Corollary 23. Let , and be given coefficient matrices in (3) over with conformable sizes. Denote Then (1)The system (3) has a solution , where and .(2)The coefficient matrices in (3) satisfy (3)(4)are equivalent statements. Under these conditions, the general solution to the system (5) can be demonstrated as where , , , , and are arbitrary matrices over

4. Cramer’s Rule of the Equation (5)

Denote . By putting , and as zero-matrices with compatible dimensions in (44)–(46), we obtain the following partial solution to (5): The following theorem gives explicit determinantal representation formulas of (75)–(77).

Theorem 24. Let , , , , , . Then the partial solution (75)–(77) to Eq.(5), , , by the components possesses the following determinantal representations: (i) where are the row vector and the column vector, is the th column of and is the th row of .(ii) where is the th row of and is such that are the column vector and the row vector, respectively. and are the th row and the th column of .(iii) where is the th column of and is the th row of . The matrix is such that and is the th row of of . The matrix is such that where and is such that and is th column of . The matrix is such that where is the th column of .(iv) or where are the column vector and the row vector, respectively. is the th column of and is the th row of .(v) where is the th column of , is such that are the column vector and the row vector, respectively. is the th row of and is the th column of .(vi) where is the th column of .(vii) where are the column vector and the row vector, respectively. and are the th row and the th column of .

Proof. (i) For the first term of (75) using the determinantal representations (22) for and (30) for , we have Denote . Due to , we havewhere and are, respectively, the unit row vector and the unit column vector whose components are except the th components which are .
If we denote by the th component of a row vector , then where . So, has the determinantal representation (81).
If we denote by the component of a column vector , then So, the other determinantal representation of is (82).
(ii) Consider the second term of (75). Due to (30) for the determinantal representation of , we have for the second multiplier , where is the th row of . By applying the determinantal representations (22) and (23) for the Moore-Penrose inverses and , respectively, and due to Lemma 19 for the first multiplier , we obtain the matrix such that are the column vector and the row vector, respectively. and are the th row and the th column of . So, we have Denote . From this denotation and the equation (114), it follows (85).
(iii) Due to Lemma 19 for the third term of (75), we have where is the row vector, is th column of , and is the column vector, is the th row of .
Construct the matrices and determined by (117) and (119), respectively. Denote , , where is the th row of , and . From these denotations and Eq. (115), it follows (88).
We can obtain another determinantal representation of as well. Denote , , where is the th column of , and . From these denotations and Eq. (115), it follows (89).
(iv) Now consider the first item of (75). Using the determinantal representations (22) for , and (30) for , we haveDenote . Then, we havewhere and are, respectively, the unit row-vector and the unit column-vector whose components are except the th components which are .
If we denote by the th component of a row vector , then where . So, has the determinantal representation (94).
If we denote by the component of a column vector , then So, the other determinantal representation of is (95).
(v) For the second term of (76) using (33) for a determinantal representation of , and similarly as in the point (iv) for , we have where is the th column of , are the column vector and the row vector, respectively. is the th row of and is the th column of . Construct the matrix , where is determined by (129). Denote the matrix . From this denotation and due to (128), we obtain (98).
(vi) For the first term of (77), , the determinantal representation (102) evidently follows due to Corollary 20.
(vii) For the second term of (77), using (22) for a determinantal representation of and (34) for a determinantal representation of , we have Denote . Since , we havewhere and are, respectively, the unit row-vector and the unit column-vector.
If we denote by the th component of a row vector , then So, has the determinantal representation (103).
If we denote by the th component of a column-vector , then So, the other determinantal representation of is (104).

5. An Example

Given the matrices, Since with , we shall find the -skew-Hermitian solution to Eq. (5) with the given matrices (137) by both methods, first, using determinantal representations of generalized inverses, and then, by obtaining Cramer’s rule.

By Theorem 13 and Corollary 15, we obtain So, It is easy to check that the consistency conditions (42) of Eq. (5) are fulfilled by these given matrices. So, Eq. (5) has -Hermitian solutions.

We find the partial solution (75)–(77) by Cramer’s rule obtained in Theorem 24. Since and (determinantal) , then, by (83),

Further, by (81), we obtain By continuing of similar computations, we get

Now, we find by (85). Since and , then Further, we construct the matrix by (86). Since , then

From this by direct multiplication, it follows that

Finally, by (85), we have

Now, we find by (89). The following algorithm of finding is given by Theorem 24.(1)Find the matrix . (2)Find the matrix . (3)By (90), construct the matrix the matrix . Since , then By similar computing, we get (4)Obtain the matrix . (5)By (93), construct the matrix . Since , then .(6)By (92), construct the matrix . Since , then , where is obtained in (144). (7)Get the matrix . (8)Finally, by (89) and due to , we have . So,

Hence, After rounding coefficients, we obtain

Now, we find by (94). First, we obtain the matrix , Since , then, by (96), So, by (94), we have Continuing in like manner yields

Further, we find by (98). The algorithm of finding is given by Theorem 24 as well. By (101), we havewhere is the first column of that is the conjugate transpose of obtained in (158). By similar computing, we get By (99), we construct the matrix . Since , then for all So, We obtain the following matrices: Finally, sincethen by (98),Hence,

By rounding coefficients, we have

Further, we find by (102). So,

Since , then by (102) By similar computing, we get

Now, we find by (103). So, Then by (103), we have Continuing in like manner yields Finally,

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

6. Conclusion

Some necessary and sufficient conditions for the existence of the general solution of the skew--Hermitian solution to (5) are constructed in this paper. We not only obtain the explicit form of the -skew-Hermitian solution in terms of generalized inverses but also give its explicit determinantal representation formulas that are an analog of Cramer’s rule. Our proposed Cramer’s rule is based on the theory of row-column noncommutative determinants. A numerical example demonstrates applying the obtained Cramer’s rule to finding the skew--Hermitian solution to the equation of (5).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interests.

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Copyright © 2019 Abdur Rehman et al. 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.

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