Closed-form formula for a classical system of matrix equations

Abstract Keeping in view the latest development of anti-Hermitian matrix in mind, we construct some closed form formula for a classical system of matrix equations having anti-Hermitian nature in this paper. We give the necessary and sufficient conditions for the existence of its solution by applying the properties of matrix rank. The general solution to this system is expressed by closed formula based on generalized inverses of given matrices. The novelty of the proposed results is not only obtaining a formal representation of the solution in terms of generalized inverses but the construction of an algorithm to find its explicit expression as well. To conduct an algorithm and numerical example, it is used the determinantal representations of the Moore Penrose inverse previously obtained by one of the authors.


Introduction and preliminaries
In the whole paper, C denotes the complex number field and the collection of all m Â n matrices over the complex number field C is denoted by C mÂn : The real number field is represented by R: The conjugate transpose matrix of A is written by A Ã and a matrix is said to be anti-Hermitian if A Ã ¼ ÀA: Both symbols r(A) and rankA stand for the rank of A. An identity matrix with feasible shape is denoted by I. The Moore-Penrose inverse of A is represented by A † ¼ U and is determined by the four Penrose's equations AUA ¼ A, UAU ¼ U, ðAUÞ Ã ¼ AU, ðUAÞ Ã ¼ UA: Furthermore, L A ¼ IÀA † A and R A ¼ IÀAA † are projectors inducted by A, and A lot of the problems in different areas from engineering like linear descriptor systems (Gavin & Bhattacharyya, 1983), system design (Syrmos & Lewis, 1994), singular system control (Darouach, 2006), perturbation theory (Li, 2000), feedback (Syrmos & Lewis, 1993), etc., to medical researches based on mathematics models with partial differential equations (Kumar, Kumar, Osman, & Samet, 2020;Osman & Machado, 2018) require solutions of Sylvester-type matrix equations. For instance, Bai carried out the iterative solution of A 1 X þ XA 2 ¼ B in Bai (2011). The consistent condition of A 1 X þ YA 2 ¼ B to have a solution was evaluated by Roth (1952) and its general solution was researched by Baksalary and Kala (1979).
Recently, the general solution of in quaternion matrices (He, Wang, & Zhang, 2018), and the general /-Hermitian solution to mixed pairs of the quaternion Sylvester matrix equations was explored in He, Liu, and Tam (2017). The study of the properties of the matrix rank is important in establishing the necessary and sufficient conditions for the existence of the solution of matrix equations.
Recently, new results on matrix ranks have been derived in Ma (2021). Some iterative algorithms of solving coupled matrix equations can be found in Chen (2005, 2006). The numerical solution of bi-sided Sylvester matrix equation was examined in Byers and Rhee (1995). The triangular bi-sided Sylvester matrix equation was researched in Jonsson and Kågstr€ om (2002). The Hermitian solutions to have explored using direct methods by simultaneous decomposition of a matrix triplet for of given complex matrices by Liu and Tian (2011), and by determinantal representations of given quaternion matrices by . Some new investigations on (1.2) were obtained in Deng and Hu (2005).
Recently, Hajarian (2015) has developed the algorithm to find out the solution of Motivated by the above findings and the remarkable usage of generalized Sylvester matrix equations in various applied areas, in this paper we explore the anti-Hermitian coupled Sylvester matrix equations over the complex number field C: Solving (1.3) will definitely enrich the usage of anti-Hermitian Sylvester matrix equation into a large number of fields. We give the necessary and sufficient conditions for the existence of its solution by applying the properties of the matrix rank. The following lemma has crucial role in gaining these results.
The principal objective of this paper is to search out the general solution to (1.3) when this system is solvable. The general solution of has important function in achieving the core result of this paper having anti-Hermitian nature over C: Lemma 1.2 (Rehman, Kyrchei, Ali, Akram, & Shakoor, 2019). Let A 4 , B 4 , C 4 , and D 4 ¼ ÀD Ã 4 be known coefficient matrices in (1.4) over C with agreeable sizes. Assume Then the terms given below are alike: (1) The system (1.4) has a solution (X, Y, Z), where Y and Z are anti-Hermitian matrices.
The anti-Hermitian solution to the system (1.3) will be expressed in terms of the Moore-Penrose (MP) inverse. The novelty of the given results is obtaining a formal representation of the solution in terms of generalized inverses and the construction of an algorithm to find its explicit expression as well. Due to the important role of generalized inverses in many application fields, considerable effort has been exerted toward the numerical algorithms for fast and accurate calculation of matrix generalized inverse. In general, most existing methods for their obtaining are iterative algorithms for approximating generalized inverses of complex matrices (some recent papers, see, e.g. Artidiello, Cordero, Torregrosa, & Vassileva, 2019;Guo & Huang, 2010;Sayevand, Pourdarvish, Machado, & Erfanifar, 2021). There are only several direct methods finding MP-inverse for an arbitrary complex matrix A 2 C mÂn : The most famous is method based on singular value decom- The computational cost of this method is dominated by the cost of computing the SVD, which is several times higher than matrix-matrix multiplication. Another approach is constructing determinantal representations of the MP-inverse A † : A well-known determinantal representation of an ordinary inverse is the adjugate matrix with the cofactors in entries. It has an important theoretical significance and bring forth Cramer's rule for systems of linear equations. The same is desirable to have for the generalized inverses. Due to looking for their more applicable explicit expressions, there are various determinantal representations of generalized inverses (for the MPinverse, see, e.g. Bapat, Bhaskara, & Prasad, 1990;Stanimirovic, 1996). Because the complexity of the previously obtained expressions of determinantal representations of the MP-inverse, they have a little applicability.
In this paper, we will used the determinantal representations of the MP-inverse recently obtained in Kyrchei (2008). (1.5) Here jAj a a denote a principal minor of A whose rows and columns are indexed by a :¼ a 1 , :::, a k f g 1, :::, m f g, Also, a Ã :j and a Ã i: denote the jth column and the ith row of A Ã , and A i: ðbÞ and A :j ðcÞ stand for the matrices obtained from A by replacing its ith row with the row vector b 2 C 1Ân and its jth column with the column vector c 2 C m , respectively.
The formulas (1.5) give very simple and elegant determinantal representations of the MP-inverse. So, for any A 2 C mÂn r , we have sum of all principal minors of r order of the matrices A Ã A or AA Ã in denominators and sum of principal minors of r order of the matrices ðA Ã AÞ :i ða Ã :j Þ or ðAA Ã Þ j: ða Ã i: Þ that contain the ith column or the jth row, respectively, in numerators into (1.5).
Note that for an arbitrary full-rank matrix A, Lemma 1.3 gives a new way of finding an inverse matrix.
ij Þ 2 C nÂm possess the following determinantal representations: Note that these new determinantal representations of the Moore-Penrose inverse have been obtained by the developed novel limit-rank method in the case of quaternion matrices (Kyrchei, 2011) as well. This method was successfully applied for constructing determinantal representations of other generalized inverses in both cases for complex and quaternion matrices (see e.g. Kyrchei, 2017aKyrchei, , 2017b. It also yields Cramer's rules of various matrix equations (Kyrchei, 2012(Kyrchei, , 2018a(Kyrchei, , 2018b(Kyrchei, , 2021Rehman, Kyrchei, Ali, Akram, & Shakoor, 2020. Our paper is composed of four sections. The general solution to (1.3) is constituted in Section 2 with a special case. The algorithm and numerical example of finding the anti-Hermitian solution to (1.3) are presented in Section 3. A conclusion to this paper is given in Section 4.

Main result
Then the following conditions are equivalent: (1) System (1.3) is consistent.
w Now we discuss the particular case of our system. If A 2 , B 2 and C 2 are all equal to zero in Theorem 2.1, then we get the following outcome.
Corollary 2.1. Given that A 1 , B 1 , and C 1 are matrices of feasible shapes over C. Assign Then the following conditions are equivalent: (1) System (1.2) is consistent. ( Under these conditions, the general solution to (1.2) can be represented as and U 3 are free matrices of feasible shapes over C:

Algorithm with example
In this section, we construct the algorithm for finding solutions to (1.3) that is inducted by Theorem 2.1.
The following example will be considered by using Algorithm 3.1. Note that our goal is both to confirm correctness of main results from Theorem 2.1 and to demonstrate the technique of applying the determinantal representations of the MP-inverse from Lemma 1.3 by using a not too complicated and understandable example.

Conclusions
A closed form formula for the anti-Hermitian solution of a classical system of matrix equations are constructed in this paper. Some viable necessary and sufficient conditions are also discussed when this system is consistent over a complex field C by applying properties of matrix rank. Special case of the researched system is also discussed. To give an algorithm finding the explicit numerical expression of the solution, it is used the determinantal representations of the MP-inverse recently obtained by one of the authors. The novelty of the conducted research is obtaining necessary and sufficient conditions to exist a solution, its formal representation of by closed formula in terms of generalized inverses, and the construction of an algorithm to find its explicit expression. A numerical example is also given to interpret the results established in this paper. It is hoped that the developed ways of obtaining necessary and sufficient conditions to existing of a solution, representation its by generalized inverses, and constructing of algorithms by using determinantal representations of generalized inverses have potential applications to solving of a wide class of matrix equations, which is an area deserving of further study.