Volume 2, 2019
Advances in Researches of Quaternion Algebras
|Number of page(s)||15|
|Section||Mathematics - Applied Mathematics|
|Published online||09 July 2019|
Cramer’s rules for the system of quaternion matrix equations with η-Hermicity
Pidstrygach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, Lviv 79060, Ukraine
* Corresponding author: firstname.lastname@example.org
Accepted: 6 June 2019
The system of two-sided quaternion matrix equations with η-Hermicity, A1XA1η* = C1, A2XA2η* = C2 is considered in the paper. Using noncommutative row-column 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 η-Hermitian solution when C1 = Cη*1 and C2 = Cη*2 and for an η-skew-Hermitian solution when C1 = −Cη*1 and C2 = −Cη*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 / η-Hermicity
© 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.
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 two-sided matrix equations
Convergence analysis in statistical signal processing and linear modeling [14, 15, 23] are some fields in which the applications of η-Hermitian matrices matrices can be viewed. The singular value decomposition of the η-Hermitian matrix was examined in . Very recently, Liu  determined η-skew-Hermitian solutions to some classical matrix equations and, among them, the generalized Sylvester-type matrix equation:
bearing η-Hermicity over by expressing it’s general η-Hermitian solution in terms of the Moore–Penrose inverses. An iterative algorithm for determining η (-skew)-Hermitian least-squares solutions to the quaternion matrix equation (3) was established in . For more related papers on η-Hermicity and its generalization, ϕ-Hermicity, 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 row-column 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 row-column determinants which were introduced in [42, 43].
Currently, applying of row-column 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 η-matrices in Section 2.1, and in the theory of row-column determinants and determinantal representations of the Moore–Penrose inverses of a quaternion matrix, its Hermitian adjoint and η-Hermitian adjoint matrices in Section 2.2. Determinantal representations of a general, η-Hermitian and η-skew-Hermitian solutions to the system (2) are derived in Section 3. Finally, the conclusion is drawn in Section 4.
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.
 Let . Then
 Let A , B and C be given matrices with right sizes over . Then
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
Similarly, elements of the main diagonal of an η 1-skew-Hermitian matrix should be as follows
Determinantal representations of generalized inverses and of solutions to some quaternion matrix equations
Through the non-commutativity 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 , 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 .
 The ith row determinant of is called by setting for all i = 1, …, n,
 The jth column determinant of is called by setting for all j = 1, …, n,
So, for a 2×2-matrix with quaternion settings , we have the four (row-column) determinants
Since for all i, j = 1, 2, they are not equal to each others, in general.
We state some properties of row-column determinants needed below.
 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
 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
 Let . Then , for all i = 1, …, n.
for all i = 1, …, n, then, due to Lemma 2.5, the next lemma follows immediately.
Let . Then
Since  for Hermitian A we have
Its properties have been completely studied in . 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,
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.
 If , then its Moore–Penrose inverse is determined as follows
For an arbitrary full-rank matrix , a row-vector , and a column-vector , 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)
If , then the Moore–Penrose inverse have the following determinantal representations:
Since , then we can use the denotation . By Lemma 2.5, for the Hermitian adjoint matrix , its Moore–Penrose inverse can be expressed as
If then its inducted projection matrices and are determined as follows
Cramer’s rule for the system (2)
The next lemma gives the explicit matrix form of a general solution to the system (1).
In that case, the general solution of (1) can be expressed as
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.
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
By putting Z, W as zero-matrices, 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
Suppose that and are the unit row and column vectors such that all their components are 0 except the lth components which are 1.
If we denote by
Similarly above, for the second term of (15),
The third term of (15) can be obtained similarly as well. So,
(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 jth column of and construct the matrix . Finally, denote . From these denotations and the equation (28), it follows
For , we have
Denote , where is determined in (29). So, similarly to the previous case, we obtain
Consider the sixth term . So,
Therefore, we proved the following theorem.
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,
But taking into account (39), we obtain
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.
Using row-column noncommutative determinants previously introduced by the author, the determinantal representations (analogs of Cramer’s rule) of the general, η-Hermitian and η-skew-Hermitian 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 η-Hermitian 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 η-Hermicity. 4open, 2, 24.
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