A Solution of System of Linear Matrix Equations by Matrix of Matrices

Sohag J. Sci. 3, No. 2, 9-17 (2018) 9 Sohag Journal of Science An International Journal

http://dx.doi.org/10.18576/sjs/030201

A solution of system of linear matrix equations by matrix of matrices

Z. Kishka1, M. Abdalla2 and A. Elrawy2,∗ 1 Dept. of Math., Faculty of Science, Sohag University, Sohag 82524, Egypt. 2 Dept. of Math., Faculty of Science, South Valley University, Qena 83523, Egypt.

Received: 2 Feb. 2018, Revised: 27 Mar. 2018, Accepted: 2 Apr. 2018 Published online: 1 May 2018

Abstract: It is well-known that the matrix equations play an important role in engineering and applied sciences. In this paper, we deﬁne and study a system of linear matrix equations. Also, we deﬁne the notions of vector of matrices (for short, VMs), linearly dependent, independent VMs and rank of the matrix of matrices (for short, MMs) which introduced recently by Kishka and et al. [1,2]. We propose three methods for computing the solution of this system.

Keywords: MMs, system of linear matrix equations, rank of MMs

1 Introduction

In recent years, several articles are written about the system of matrix equations. Mitra [3], ﬁrst study the system

A1XB1 = C1, A2XB2 = C2, over the complex number C. Many authors contributed to this system (see [4,5,6,7,8,9,10]). To instance, in 1987, Van der Woude investigated this system over a field. In 1996, Wang studied the system over an arbitrary division ring. Kiska et al. (cf. [1,2]) introduced the concept of MMs which provide the possibility of extending the many topics of matrices in addition to saving time and speed in the work of many mathematical calculations. Our fundamental outcomes, define a linear system of matrix equations, and we introduce several methods to solve this system. The remainder of this article is organized as follows: In section 2, we present the notions of VMs, linearly dependent, linearly independent VMs and rank of MMs. Section 3, is devoted to show some methods for computing the solution of linear system of matrix equations. The conclusion is reported in section 4. Throughout this paper, the following notations are used: K K K ′ K K Let be an arbitrary field and Ml ( ) be the set of all l × l matrices over , Ml ( ) ⊂ Ml ( ) be commutative ring I and O stand for the identity matrix and the zero matrix in Ml (K), respectively. We denote by Mm×n (Ml (K)) the set of all m × n MMs over Ml (K), the elements of Mm×n (Ml (K)) are denoted by A , B, C , D, G , ...etc. Now, it is useful to define MMs and the system of linear matrix equations.

Deﬁnition 1.[2] Let A ∈ Mm×n (Ml (K)), then it can be written as a rectangular table of elements Ai j; i = 1,2,...,m and j = 1,2,...,n, as follows: A11 ... A1n A  . . .  = . .. . .  Am1 ... Amn    ∗ Corresponding author e-mail: [email protected]

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The system of m matrix equations in n unknowns deﬁned as follows:

A11X1 + ...... + A1nXn = B1, A21X1 + ...... + A2nXn = B2, . (1.1) . Am1X1 + ..... + AmnXn = Bm, where Ai j, Xi and B j ∈ Ml (K), i = 1,2,...,n, j = 1,2,...,m, and Ai j are called the coefﬁcients, Xi are variables, and B j are scalars matrices called the constants. We can write the above system (1.1) in the form A X = B, (1.2) by using the concept MMs.

Remark.It is said that system of matrix equations that homogeneous if Bi = O, where O is zero matrix. Otherwise called non-homogeneous. Deﬁnition 2.Two systems of matrices equations involving the same matrices of variables are said to be equivalent if they have the same solution set. Theorem 1.Any system of matrix equations has one of the following exclusive conclusions: (1) No solution. (2) Unique solution. (3) Inﬁnitely many solutions. Before presenting solution of the system of the linear matrix equations, it is useful to introduce a few notations.

2 Basic points

In this section, we introduce a deﬁnition for VMs, linearly dependent, linearly independent VMs and rank of MMs. Also, we show the type of solution.

Deﬁnition 3.Let Ml (K) be a ring with unity, then the set n V = {(A1,A2,...,An) : A1,A2,...,An ∈ Ml (K)}, is called VMs. Deﬁnition 4.The set of n VMs is linearly independent if n ∑ EkVk = V0, k=1 and Ek = O, where Vk = Ak j , V0 = O1,k ∈ M1×n (Ml (K)), O, Ek ∈ Ml (K) and k = 1,2,...,n. If oneofEk 6= O, then n VMs is linearly dependent.

Deﬁnition 5.Let A = (Ai j) ∈ Mm×n (Ml (K)), then

rank (A )= max {rank A1 j } + ... + max {rank (Ap j)} 1≤ j≤n 1≤ j≤n p = ∑ max {rank (Ai j)}, 1 j n i=1 ≤ ≤ where p denoted the number of linear independent rows. If the augmented MMs (A |B) is obtained from (1.2), then we have the following properties (1)If rank(A )= rank(A |B)= ρ, then the system has a unique solution. (2)If rank(A )= rank(A |B) < ρ, then the system has ∞−many solutions. (3)If rank(A ) < rank(A |B), then the system has no solution. Where ρ is the number of rows in X product size of elements.

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3 Solve the system of the linear matrix equations

In this section, we introduce some methods to solve the system of the linear matrix equations.

3.1 The inverse MMs method

In this subsection, we use the inverse of MMs to solve the system of the linear matrix equations. ′ K A M ′ K B M ′ K We consider Ml ( ) be commutative matrices, and ∈ n Ml ( ) , ∈ n×1 Ml ( ) and X M ′ K ∈ n×1 Ml ( [x]) . Furthermore, we consider the system, A X = B, then the solution of this system is X = A −1B.

Example 1.Let 23 13 X1 + X2 = I2, 13 12

26 2 −3 X1 + X2 = O2, 24 −1 1

x1 x2 y1 y2 ′ K where X1 = , X2 = , I2 and O2 ∈ M2 ( ), then x3 x4 y3 y4

rank(A )= rank(A |B)= 4, so the system has a unique solution. Since the inverse A

9 1 1 − 7 7 0 − 3 4 0 1 A −1 =  7 7 7 , 2 0 1 − 3  7 7 7   0 2 − 1 0   7 7  then the solution is 9 1 − 7  − 3 4  X = 7 7 . 2 0  7   0 2   7  Example 2.Let 123 10 −1 1  204 X1 +  0 0 10 X2 = O3, 231 0 8 2     12 −1 1  0 3 10 X1 + I3X2 = I3, 0 8 4   x11 x12 x13 y11 y12 y13 ′ K where X1 =  x21 x22 x23 , X2 =  y21 y22 y23 , I3 and O3 ∈ M3 ( ), then x31 x32 x33 y31 y32 y33     rank(A )= rank(A |B)= 6,

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so the system has a unique solution. Since the inverse A

1407 17 83 7035 109 714 − 167276 − 17608 167276 83638 − 8804 41819 51 545 2143 255 905 11885   − 334552 − 35216 334552   167276 − 17608 83638   − 43 231 − 4753 215 1999 − 3099 A −1 =   334552 35216 334552   167276 17608 83638    4221 − 167 1345 − 391 −135 − 1197   41819 17608 83638 41819 8804 41819   153 − 175 21627 − 765 333 − 2340    83638 35216 167276   41819 17608 41819    129 3305 − 1445 − 645 − 755 487    83638 35216 167276   41819 17608 41819   then the solution is 7035 109 714 83638 − 8804 41819 255 905 11885   167276 − 17608 83638   215 1999 − 3099 X =  167276 17608 83638  .  − 391 −135 − 1197   41819 8804 41819   − 765 333 − 2340    41819 17608 41819    − 645 − 755 487    41819 17608 41819  

3.2 Gaussian elimination on MMs

Gaussian elimination method is the standard method for solving linear equations (see [11]). In this subsection, we offer Gaussian elimination method in the case of MMs to solve the system of the linear matrix equations. Now, we introduce elementary row operations and row echelon forms. Deﬁnition 6.There are three kinds of elementary row operations on matrices: (a)Adding a multiple of one row to another row; (b)Multiplying all entries of one row by a nonzero matrix constant; (c)Interchanging two rows.

Definition 7.A MMs is in row echelon form when it satisfies the following conditions: (i)The first non-zero matrix element in each row, called the leading entry, is identity matrix. (ii)Each leading entry is in a column to the right of the leading entry in the previous row. (iii)Rows with all zero matrix elements, if any, are below rows having a non-zero matrix element.

Definition 8.A MMs is in reduced row echelon form when it satisfies the following conditions: (i)The MMs is in row echelon form (i.e., it satisfies the three conditions listed above). (ii)The leading entry in each row is the only non-zero matrix entry in its column.

Example 3.Applying the Gaussian elimination method on the same Example 1, it follows that

23 13 . . I2 −1  13 12  23 r1 26 2 −3 . 13  . O2 −−−−−−−−→  24 −1 1   

0 1 . 1 −1 I2 1 2 . 1 2  3 3 − 3 3  26 − r1 + r2 26 2 −3 . 24  . O2 −−−−−−−−−−−→  24 −1 1    0 1 . 1 −1 I2 1 2 . 1 2 −1  3 3 − 3 3  0 −7 . −7 −7 r2 0 −7 . 0 −2 3 3  O2 .   −7 −7 − 2 − 2 −−−−−−−−−−→  3 3 3 3 

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0 1 . 1 −1 I2 1 2 . 1 2 −1  3 3 − 3 3  0 1 . 2 − −1 2 r2 + r1 . 7 0 3 3  O2 I2 .   0 2 −−−−−−−−−−−−−−→  7  . 1 − 9 I O . 7 2 2 − 3 4  7 7 , . 2 0  O I . 7   2 2 0 2   7  9 2 1 − 7 7 0 so A = 3 4 and B = 2 . − 7 7 0 7 Example 4.Let 23 13 X1 + 3I2X2 + X3 = I2, 13 12 26 33 2 −3 X1 + X2 + X3 = 2I2, 24 16 −1 1 39 16 X1 + X2 + 2I2X3 = O2, 36 23

x1 x2 y1 y2 z1 z2 where X1 = , X2 = , X3 = , I2 and O2 ∈ M2 (K), then x3 x4 y3 y4 z3 z4

23 13 . 3I2 . I2  13 12  −1 26 33 2 −3 . 23  .2I2  r1  24 16 −1 1  13  −−−−−−−−→  39 16 .   2I2 . O2   36 23    3 −3 0 1 . 1 −1 I2 1 1 . −1 2  −1 2 3 3 3 3  26 − r1 + r2& 26 33 2 −3 . 24  . 2I2   24 16 −1 1  39   − r1 + r3  39 16 .  36  2I2 . O2 −−−−−−−−−−−−−→  36 23    3 −3 0 1 . 1 −1 I2 .  −1 2 1 1 −1 2  3 3 3 3 −1 3 −3 0 −7 . 4 −6 3 −3  O2 .  r2  −1 4 − 7 − 7 4 − 8  −1 4  3 3 3 3 −−−−−−−−−−→  1 −3 −1 −6 . 3 −9   O2 .   −1 0 −2 −3 2 −7    3 −3 0 1 . 1 −1 I2 1 1 . −1 2  −1 2 3 3 3 3  3 −3 7 35 . 20 32 − r2 + r1& − 9 − 9 . 9 − 9 −1 2  O2 I2 7 14 . 8 14   − 9 − 9 9 9  1 −3   − r2 + r2  1 −3 −1 −6 . 3 −9  −1 0  O .   2 −1 0 −2 −3 2 −7 −−−−−−−−−−−−−−−−→   0 8 . −4 7 I2 O2 10 −4 . 1 2  9 9 9 9  7 35 . 20 32 5 16 −1 − 9 − 9 . 9 − 9 − 9 − 9  O2 I2 7 14 . 8 14  11 34 r3  − 9 − 9 9 9  − 9 − 9  5 16 . 31 91 −−−−−−−−−−−−→  − 9 − 9 . 9 − 9   O2 O2 .   − 11 − 34 − 38 − 95   9 9 9 9 

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0 8 . −4 7 I2 O2 10 −4 . 1 2  9 9 9 9  0 8 7 35 . 20 32 − 10 −4 r3 + r1& − 9 − 9 . 9 − 9 9 9  O2 I2 7 14 . 8 14  7 35  − 9 − 9 9 9  − 9 − 9  . 223 787  − 7 14 r3 + r2  . 3 − 3  − 9 − 9  O2 O2 I2 .   − 151 263 −−−−−−−−−−−−−−−→  9 3  . 592 2083 . 3 − 3 I2 O2 O2 . 281 992  − 3 3  . 227 400 . − 6 3  O2 I2 O2 . ,  176 − 623   9 9   . 223 − 787   O O I . 3 3   2 2 2 − 151 263   9 3  592 2083 227 400 223 787 3 − 3 − 6 3 3 − 3 so X1 = 281 992 , X2 = 176 623 and X3 = 151 263 . − 3 3 9 − 9 − 9 3 Example 5.Let

I2X1 + 2 I2X2 = O2,

3 I2X1 + 4 I2X2 = O2,

x1 x2 y1 y2 where X1 = , X2 = , I2 and O2 ∈ M2 (K), then the system has ∞−many solutions. x3 x4 y3 y4 Example 6.Let 10 12 I2X1 + X2 + I2X3 = , 00 21 12 13 X1 + O2X2 + 2I2X3 = , 21 31

x1 x2 y1 y2 z1 z2 where X1 = , X2 = , X3 = , I2 and O2 ∈ M2 (K), then the system has no solution. x3 x4 y3 y4 z3 z4

3.3 Crammer’s rule on MMs

In this subsection, we give Crammer’s rule on MMs and use them in solving system of linear matrix equations. ′ K K A M ′ K Throughout this subsection, we consider Ml ( ) ⊂ Ml ( ) be commutative matrices, and ∈ n Ml ( ) , B M ′ K X M ′ K A A ∈ n×1 Ml ( ) and ∈ n×1 Ml ( (x)) and let i be the MMs obtained from by replacing the ith column of A by the column vector B. Furthermore, let

D = detA , N1 = detA1,..., Nn = detAn. The fundamental relationship between determinants and the solution of the system A X = B will be given in the following theorem. Theorem 2.The square system A X = B has a solution if and only if D is non-singular.In this case, the unique solution is given by −1 Xi = D Ni, i ∈{1,2,...n}. Proof.The (square) system A X = B has a unique solution if and only if A is invertible, and A is invertible if and only if D = detA is non-singular. Now suppose D non-singular, then A −1 = D−1ad jA . Multiplying A X = B by A −1, we obtain X = A −1A X =D−1 (ad jA )B. (3.1) 1 A 1 A A A B t Note that the ith row of D (ad j ) is D ( 1i, 2i,..., ni). If =(B1,B2,...,Bn) , then by (3.1),

−1 Xi = D (B1A1i + B2A2i + ... + BnAni).

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However, B1A1i + B2A2i + ... + BnAni = Ni, the determinant of the matrix obtained by replacing the ith column of A by the column vector B. Thus, −1 Xi = D Ni.

Example 7.Applied Theorem 2 in Example 1

23 13 X1 + X2 = I2, 13 12

26 2 −3 X1 + X2 = O2, 24 −1 1

x1 x2 y1 y2 where X1 = , X2 = , then x3 x4 y3 y4

23 13  13 12  −7 −21 D = detA = det = , 26 2 −3 −7 −14    24 −1 1    and

13 I2  12  2 −3 NX = det = , 1 2 −3 −1 1  O   2 −1 1    also

23 I2  13  26 NX = det = − , 2 26 24  O2   24    the unique solution is given by

9 2 −1 1 − 7 −1 7 0 X1 = D NX1 = 3 4 , X2 = D NX2 = 2 . − 7 7 0 7

4 Conclusion

In this work, we have introduced a deﬁnition for VMs, linearly dependent, independent VMs and rank of the MMs. Also, the system of the linear matrix equations are solved by using some methods. It is worthy to ensure that when we write the above matrix linear equations in the case of systems of linear equations whose coefﬁcients are the scalar number instead of matrices and solve any system by using the classical methods we shall obtain the same results but we shall need a lot of papers and time.

Acknowledgment

The authors are very grateful to the anonymous referees for many valuable comments and suggestions which helped to improve the paper.

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References

[1] Z. Kishka, M. Saleem, S. Sharqawy and A. Elrawy, On matrix of matrices over semirings, Math. Sci. Lett., 7(2), (2018), 107-110. [2] Z. Kishka, M. Saleem, M. Abdalla and A. Elrawy, On Hadamard and Kronecker products over matrix of matrices, General Letters in Mat., 4(1), (2018), 13-22. [3] S.K. Mitra, A pair of simultaneous linear matrix equations A1XB1 = C1and A2XB2 = C2, Proc. Cambridge Philos. Soc. 74, (1973), 213–216. [4] A.B. Ozg¨uler and N. Akar, A common solution to a pair of linear matrix equations over a principle domain, Linear Algebra Appl. 144, (1991), 85-99. [5] A. Navarra, P.L. Odell and D.M. Young, A representation of the general common solution to the matrix equations A1XB1 = C1and A2XB2 = C2 with applications, Comput. Math. Appl. 41, (2001), 929–935. [6] J. Van der Woulde, Almost noninteracting control by measurement feedback, Systems Control Lett.9, (1987), 7–16. [7] J. Van der Woulde, Feedback Decoupling and Stabilization for Linear System with Multiple Exogenous Variables, Ph.D. Thesis, Technical University of Eindhoven, Netherlands, (1987). [8] N. Shinozaki and M. Sibuya, Consistency of a pair of matrix equations with an application, Keio Eng. Rep. 27, (1974), 141–146. [9] Q.W. Wang, The decomposition of pairwise matrices and matrix equations over an arbitrary skew ﬁeld, Acta Math. Sinica 39, (1996), 396–403. [10]S.K. Mitra, A pair of simultaneous linear matrix equations and a matrix programming problem, Linear Algebra Appl. 131, (1990), 97–123. [11]S. Gharib, S. R. Ali, R. Khan, N. Munir and M. Khanam, System of linear equations, Gaussian elimination, Global J. Inc., (2015).

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Zanhom M. Kishka has got Ph.D. degree from Assiut University 1984 in the ﬁeld of complex analysis. He is professor in mathematics department, faculty of science, Sohag University. His research interests are in the areas complex analysis and theory of matrices. He has published research articles in reputed international journals of mathematical.

Mohamed. A. Abul-Dahab earned his M.Sc. degree. In 2010 in pure mathematics from South Valley University, faculty of science, Qena, Egypt. In 2013, he received his Ph.D. degree from South Valley University, faculty of science, Qena, Egypt. His research interests include complex analysis, special function, and matrix analysis. He has published research articles in reputed international journals of mathematical sciences. He is referee and editor of mathematical journals.

Amr M. Elrawy has M.Sc. degree in mathematics science from South Valley University 2013. He assistant lecturer in mathematics department, faculty of science, South Vallely University. His research interests are in the areas of algebraic geometry and algebra. He has published research articles in reputed international journals of mathematical.

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