Review Article
Cramer’s rules for the system of quaternion matrix equations with ηHermicity
Pidstrygach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, Lviv 79060, Ukraine
^{*} Corresponding author: st260664@gmail.com
Received:
10
January
2019
Accepted:
6
June
2019
The system of twosided quaternion matrix equations with ηHermicity, A_{1}XA_{1}^{η*} = C_{1}, A_{2}XA_{2}^{η*} = C_{2} is considered in the paper. Using noncommutative rowcolumn determinants previously introduced by the author, determinantal representations (analogs of Cramer’s rule) of a general solution to the system are obtained. As special cases, Cramer’s rules for an ηHermitian solution when C_{1} = C^{η*}_{1} and C_{2} = C^{η*}_{2} and for an ηskewHermitian solution when C_{1} = −C^{η*}_{1} and C_{2} = −C^{η*}_{2} are also explored.
Mathematics Subject Classification: 15A24 / 15A15 / 15A09 / 15B33
Key words: Generalized inverse / Noncommutative determinant / Quaternion matrix / System of matrix equations / Cramer rule / ηHermicity
© I.I. Kyrchei, Published by EDP Sciences, 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Introduction
In the whole article, the notation is reserved for the real number field and stands for the set of all m × n matrices over the quaternion skew field
specifies its subset of matrices with a rank r. For given , the conjugate of h is . For given , A* represents the conjugate transpose (Hermitian adjoint) matrix of A. The matrix is Hermitian if A* = A. A ^{†} means the Moore–Penrose inverse of , i.e. the exclusive matrix X satisfying the following four equations
Quaternions have ample use in diverse areas such, such as color imaging and computer science [1–5], fluid mechanics [6, 7], quantum mechanics [8, 9], the attitude orientation and spatial rigid body dynamics [10–12], signal processing [13–15], etc.
The research of matrix equations have both applied and theoretical importance. Many authors explored the system of twosided matrix equations
(1)over the field of complex numbers, the quaternion skew field, etc. (see, e.g. [16–21]). In this paper, the following system of quaternion matrix equations with ηHermicity are considered,
Definition 1.1.
[22–24] A matrix is known to be ηHermitian and ηskewHermitian if and , respectively, where .
Convergence analysis in statistical signal processing and linear modeling [14, 15, 23] are some fields in which the applications of ηHermitian matrices matrices can be viewed. The singular value decomposition of the ηHermitian matrix was examined in [22]. Very recently, Liu [25] determined ηskewHermitian solutions to some classical matrix equations and, among them, the generalized Sylvestertype matrix equation:
(3)Note that in [25], the term “ηantiHermitian” has been used instead “ηskewHermitian”. He and Wang [26] gave the general solution of
bearing ηHermicity over by expressing it’s general ηHermitian solution in terms of the Moore–Penrose inverses. An iterative algorithm for determining η (skew)Hermitian leastsquares solutions to the quaternion matrix equation (3) was established in [27]. For more related papers on ηHermicity and its generalization, ϕHermicity, one may refer to [28–38].
In this paper, we construct novel explicit determinantal representation formulas (an analog of Cramer’s rule) of the general and η(skew)Hermitian solutions to the system (2), by using determinantal representations of the Moore–Penrose matrix that was obtained within in the framework of the theory of rowcolumn noncommutative determinants. According to our best of knowledge, our Cramer’s rule proposed is a unique direct method to compute the η(skew)Hermitian solutions to quaternion matrix equations unlike other similar works (see, e.g. [24–26, 29, 32]), where obtained explicit forms of solutions have mostly only theoretical significance.
In contrast to the inverse matrix that has a definitely determinantal representation in terms of cofactors, for generalized inverse matrices, in particular, Moore–Penrose matrices, there exist different determinantal representations even for matrices with real or complex entries as a result of the search of their more applicable explicit expressions (for the Moore–Penrose matrix, see, e.g., [39–41]). For quaternion matrices, in view of the noncommutativity of quaternions, the problem of the determinantal representation of generalized inverse matrices remained open for a long time and only now can be solved due to the theory of rowcolumn determinants which were introduced in [42, 43].
Currently, applying of rowcolumn determinants to determinantal representations of various generalized inverses have been derived by the author (see, e.g. [44–57]) and other researchers (see, e.g. [58–61]). In particular, determinantal representations of systems like to (1) have been recently explored in [53, 55, 56, 61].
The remainder of the paper is directed as follows. In Section 2, we start with preliminaries in general properties generalized inverses, projectors, and ηmatrices in Section 2.1, and in the theory of rowcolumn determinants and determinantal representations of the Moore–Penrose inverses of a quaternion matrix, its Hermitian adjoint and ηHermitian adjoint matrices in Section 2.2. Determinantal representations of a general, ηHermitian and ηskewHermitian solutions to the system (2) are derived in Section 3. Finally, the conclusion is drawn in Section 4.
Preliminaries: Determinantal representations of solutions to quaternion matrix equations
General properties generalized inverses, projectors, and ηmatrices
We begin with some famous results on generalized inverses and projectors inducted by them which will be used in the remaining part of this paper.
Lemma 2.1.
[26] Let . Then
Lemma 2.2.
[71] Let A , B and C be given matrices with right sizes over . Then
Remark 2.1.
For any for all l = 1, 2, 3, and q = q _{0} + q _{1} η _{1} + q _{2} η _{2} + q _{3} η _{3}, we denote
So, elements of the main diagonal of an η _{1}Hermitian matrix should be as follows
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 η _{1}skewHermitian matrix should be as follows
and a pair of elements which are symmetric with respect to the main diagonal can be represented as
Determinantal representations of generalized inverses and of solutions to some quaternion matrix equations
Through the noncommutativity of the quaternion skew field, determining of the determinant with noncommutative entries (it is also called a noncommutative determinant) is not so trivial (see, e.g. [62, 63]). There are several versions of the definition of noncommutative determinants (see, e.g., [64–69]). But, it is proved in [70], if all functional properties of determinant over a ring are satisfied, then it takes on a value in its commutative subset only. In particular, it means that such determinant can not be expanded by cofactors along an arbitrary row or column. To avoid these difficulties, for , we define n row determinants and n column determinants which are not owning of all functional properties that could be inherent to the usual determinant.
Suppose S _{ n } is the symmetric group on the set .
Definition 2.2.
[42] The ith row determinant of is called by setting for all i = 1, …, n,
where σ is the leftordered permutation. It means that its first cycle from the left starts with i, 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 2.3.
[42] The jth column determinant of is called by setting for all j = 1, …, n,
where τ is the rightordered permutation. It means that its first cycle from the right starts with j, 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, .
Remark 2.4.
So, for a 2×2matrix with quaternion settings , we have the four (rowcolumn) determinants
Since for all i, j = 1, 2, they are not equal to each others, in general.
We state some properties of rowcolumn determinants needed below.
Lemma 2.3.
[42] If the ith row of is a left linear combination of other row vectors, i.e. , where and for all l = 1, …, k and i = 1, …, n, then
Lemma 2.4.
[42] If the jth column of is a right linear combination of other column vectors, i.e. , where and for all l = 1, …, k and j = 1, …, n, then
Lemma 2.5.
[43] Let . Then , for all i = 1, …, n.
Since by Definitions 2.2 and 2.3 for
for all i = 1, …, n, then, due to Lemma 2.5, the next lemma follows immediately.
Lemma 2.6.
Let . Then
Remark 2.5.
Since [42] for Hermitian A we have
the determinant of a Hermitian matrix is called by setting for any i = 1, …, n.
Its properties have been completely studied in [43]. In particular, from them it follows the definition of the determinantal rank of a quaternion matrix A as the largest possible size of nonzero principal minors of its corresponding Hermitian matrices, i.e. .
For determinantal representations of the Moore–Penrose inverse, we use the following notations. Let and be subsets with . By denote a submatrix of with rows and columns indexed by α and β, respectively. Then, is a principal submatrix of A with rows and columns indexed by α. Moreover, for Hermitian A, is the principal minor of det A. Suppose that,
stands for the collection of strictly increasing sequences of 1 ≤ k ≤ n integers chosen from {1, …, n}. For fixed and , put , .
By and , and denote the jth columns and the ith rows of A and A*, respectively. Suppose and stand for the matrices obtained from A by replacing its ith row with the row b and its jth column with the column c, respectively.
Theorem 2.6.
[44] If , then its Moore–Penrose inverse is determined as follows
Remark 2.7.
For an arbitrary fullrank matrix , a rowvector , and a columnvector , we assume that for all i = 1, …, m, j = 1, …, n,

if rank A = n, then in (4)

if rank A = m, then in (5)
Corollary 2.1.
If , then the Moore–Penrose inverse have the following determinantal representations:
Remark 2.8.
Since , then we can use the denotation . By Lemma 2.5, for the Hermitian adjoint matrix , its Moore–Penrose inverse can be expressed as
Remark 2.9.
Suppose . By Lemma 2.6 and Remark 2.8, for the ηHermitian adjoint matrix and ηskewHermitian adjoint matrix , determinantal representations of their Moore–Penrose inverses and are respectively
Since the projection matrices and are Hermitian, then and for all i ≠ j. From Theorem 2.6 and Remark 2.8, it follows evidently the corollaries.
Corollary 2.2.
If then its inducted projection matrices and are determined as follows
(9) where and , and are the jth columns and ith rows of and , respectively.
Cramer’s rule for the system (2)
The next lemma gives the explicit matrix form of a general solution to the system (1).
Lemma 3.1.
[21] Suppose that , , , , , are known and is unknown. Put , , , . Then the system (1) is consistent if and only if
In that case, the general solution of (1) can be expressed as
(10) where Z and W are arbitrary matrices over with appropriate sizes.
Some simplification of (10) can be derived due to Lemma 2.2. So, we have,
Substituting (11) in (10), we get
By putting Z = W = 0, we get the following expression of the partial solution
Now consider the system (2). We have
similarly, , and, by Lemma 2.1, , and for i = 1, 2. Moreover, by substituting , we obtain
From above, it follows the next analog of Lemma 3.1.
Lemma 3.2.
Suppose that , , , are known and is unknown. The system (2) is consistent if and only if
In that case, the general solution to (2) is expressed as
where Z and W are arbitrary matrices over with appropriate sizes.
By putting Z, W as zeromatrices, the partial solution to (2) is
Further, we give determinantal representations of (15).
Suppose that , , , , , and . So, , , , , , and .
Consider each summand of (15) separately.

Denote . For the first term of (15) , we have
Taking into account (4) and (6) for and , respectively, we get
Suppose that and are the unit row and column vectors such that all their components are 0 except the lth components which are 1.
Since then
By
(17)denote the sth component of a rowvector . Then
Farther, it is evident that . Integrating (17) and (18) in (16), the determinantal representation of the first term of (15) can be expressed as
If we denote by
(21)the fth component of a columnvector , then
Integrating (21) and (22) in (16), we obtain another determinantal representation of the first term
are the column vector and is the fth row of .

Similarly above, for the second term of (15),
we have
Here and are the qth row and the lth column of .

The third term of (15) can be obtained similarly as well. So,
Here , are the qth row and the lth column of .

Now, consider the fourth term of (15). Taking into account (4) for determinantal representations of H ^{†} and T ^{†}, we get
(28)Here , denote the ith columns of and , respectively. is the (zf)th element of the first term that is obtained in the point (i). q _{fj} is the (fj)th element of that, by (8), can be expessed as
where is the fth row of . Denote
(29)where is the zth row of for all z, j = 1,…, n and X _{1} is found in the point (i). Construct the matrix . Further, denote
where is the jth column of and construct the matrix . Finally, denote . From these denotations and the equation (28), it follows
(30)where is the jth column of .

For , we have
Denote , where is determined in (29). So, similarly to the previous case, we obtain
(31)where is the jth column of .

Consider the sixth term . So,
Here is the lth column of and is the qth row of . Construct the matrix such that ϕ _{ qj } is determined in (33) and denote . From this denotation and the equation (32), it follows
(34)where is the jth column of .
(35)where , are the qth column of and the fth row of , respectively. Denote
(36)where is the qth row of . Construct the matrix such that ω _{qj} is determined in (36) and denote . From these denotations and the equation (35), it follows
(37)where is the jth column of .
Therefore, we proved the following theorem.
Theorem 3.1.
Suppose that , , and , . The partial solution (15) to the system (2) by components is
where the summand has the determinantal representations (19) and (23), – (24) and (25), – (26) and (27), – (30), – (31), – (34), and – (37).
Remark 3.2.
Theorem 3.1 gives the direct method of finding of a general solution to the system (2) that is an analog of Cramer’s rule. For this we need in ranks of given matrices and satisfying by them the consistent conditions (13)–(14).Let, now,
The partial ηHermitian solution Y _{1} and ηskewHermitian solution Y _{2} to the system (2) with the restrictions (38) and (39), respectively, can be expressed as
where X is an arbitrary solution to the system (2). Due to the expression of the general solution (15) and taking into account (38), we have
But taking into account (39), we obtain
So, the partial ηHermitian and ηskewHermitian solutions to the system (2) with the restrictions (38) and (39), respectively, have the following expression:
The determinantal representations of and can be obtained by components respectively as
for all i, j = 1, …, n, where x _{ ij } is determined by Theorem 3.1.
Conclusion
Using rowcolumn noncommutative determinants previously introduced by the author, the determinantal representations (analogs of Cramer’s rule) of the general, ηHermitian and ηskewHermitian solutions to the system of quaternion matrix equations , have been derived. For these purposes, determinantal representations of the Moore–Penrose inverses of a quaternion matrix, its Hermitian adjoint and ηHermitian adjoint matrices have been explored and used.
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Cite this article as: Kyrchei II 2019. Cramer’s rules for the system of quaternion matrix equations with ηHermicity. 4open, 2, 24.