A note on solutions of linear systems

A note on solutions of linear systems Branko Malešević a, Ivana Jovović a, Milica Makragić a, Biljana Radičić b arxiv:134789v2 mathra 21 Jul 213 Abstract a Faculty of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11 Belgrade, Serbia b Faculty of Civil Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11 Belgrade, Serbia In this paper we will consider Rohde s general form of {1}-inverse of a matrix A The necessary and sufficient condition for consistency of a linear system Ax = c will be represented We will also be concerned with the minimal number of free parameters in Penrose s formula x = A (1) c + (I A (1) A)y for obtaining the general solution of the linear system This results will be applied for finding the general solution of various homogenous and non- -homogenous linear systems as well as for different types of matrix equations Keywords: Generalized inverses, linear systems, matrix equations 1 Introduction In this paper we consider non-homogeneous linear system in n variables Ax = c, (1) where A is an m n matrix over the field C of rank a and c is an m 1 matrix over C The set of all m n matrices over the complex field C will be denoted by C m n, m,n N The set of all m n matrices over the complex field C of rank a will be denoted by C m n a For simplicity of notation, we will write A i (A j ) for the i th row (the j th column) of the matrix A C m n Email addresses: Branko Malešević <malesevic@etfrs>, Ivana Jovović <ivana@etfrs>, Milica Makragić <milicamakragic@etfrs>, Biljana Radičić <biljana radicic@yahoocom> Preprint submitted to ISRN Algebra April 29 213

Any matrix X satisfying the equality AXA = A is called {1}-inverse of A and is denoted by A (1) The set of all {1}-inverses of the matrix A is denoted by A{1} It can be shown that A{1} is not empty If the n n matrix A is invertible, then the equation AXA = A has exactly one solution A 1, so the only {1}-inverse of the matrix A is its inverse A 1, ie A{1}={A 1 } Otherwise, {1}-inverse of the matrix A is not uniquely determined For more informations about {1}-inverses and various generalized inverses we recommend ABen-Israel and TNE Greville1 and SL Campbell and CD Meyer 2 For each matrix A C m n a there are regular matrices P C n n and Q C m m such that QAP = E a = Ia, (2) where I a is a a identity matrix It can be easily seen that every {1}-inverse of the matrix A can be represented in the form A (1) Ia U = P Q (3) V W where U = u ij, V = v ij and W = w ij are arbitrary matrices of corresponding dimensions a (m a), (n a) a and (n a) (m a) with mutually independent entries, see C Rohde 8 and V Perić 7 We will generalize the results of N S Urquhart 9 Firstly, we explore the minimal numbers of free parameters in Penrose s formula x = A (1) c+(i A (1) A)y for obtaining the general solution of the system (1) Then, we consider relations among the elements of A (1) to obtain the general solution in the form x = A (1) c of the system (1) for c This construction has previously been used by B Malešević and B Radičić 3 (see also 4 and 5) At the end of this paper we will give an application of this results to the matrix equation AXB = C 2 The main result In this section we indicate how technique of an {1}-inverse may be used to obtain the necessary and sufficient condition for an existence of a general solution of a non-homogeneous linear system 2

Lemma 21 The non-homogeneous linear system (1) has a solution if and only if the last m a coordinates of the vector c = Qc are zeros, where Q C m m is regular matrix such that (2) holds Proof: The proof follows immediately from Kroneker Capelli theorem We provide a new proof of the lemma by using the {1}-inverse of the system matrix A The system (1) has a solution if and only if c = AA (1) c, see R Penrose 6 Since A (1) is described by the equation (3), it follows that AA (1) = AP Ia U V W Hence, we have the following equivalences c = AA (1) c c = c a c n a (I AA (1) )c= Q 1 U I n a U I n a c n a = Q = Q 1 Ia U c a Qc }{{} c n a Furthermore, we conclude c = AA (1) c c n a = Theorem 22 The vector x = A (1) c+(i A (1) A)y, c Q ( Q 1 Q Q 1 Ia U U = I n a c n a ) Q c= c = Uc = n a = y C n 1 is an arbitrary column, is the general solution of the system (1), if and only if the {1}-inverse A (1) of the system matrix A has the form (3) for arbitrary matrices U and W and the rows of the matrix V(c a y a )+y (n a) are free parameters, where Qc = c = c a y and P 1 y = y = a y n a Proof: Since {1}-inverse A (1) of the matrix A has the form (3), the solution of the system x = A (1) c+(i A (1) A)y can be represented in the form ( ) Ia U Ia U x = P Qc+ I P QA y V W V W ( ) Ia U = P c V W Ia U + I P QAPP V W 1 y 3

According to Lemma 21 and from (2) we have ( Ia U c x = P a Ia U + I P V W V W Ia P 1 ) y Furthermore, we obtain c x = P a c = P a c = P a c = P a Vc a Vc a Vc a Vc a ( Ia + I P V ( + PP 1 Ia P V ( Ia +P I V y +P a V I n a P 1 ) ya y n a P 1 ) ya ) P 1 ya y n a, y n a y n a where y = P 1 y We now conclude ( ) c x = P a c Vc + a Vy a = P a +y n a V(c a y a )+y n a c Therefore, since matrix P is regular we deduce that P a V(c a y a )+y n a is the general solution of the system (1) if and only if the rows of the matrix V(c a y a )+y n a are n a free parameters Corollary 23 The vector x = (I A (1) A)y, y C n 1 is an arbitrary column, is the general solution of the homogeneous linear system Ax =, A C m n, if and only if the {1}-inverse A (1) of the system matrix A has the form (3) for arbitrary matrices U and W and the rows of the matrix Vy a + y (n a) are free parameters, where P 1 y = y = y a y n a Example 24 Consider the homogeneous linear system x 1 +2x 2 +3x 3 = 4x 1 +5x 2 +6x 3 = (4) 4

The system matrix is For regular matrices Q = 1 4 1 A = 1 2 3 4 5 6 and P = 1 2 3 1 1 3 2 1 equality (2) holds Rohde s general {1}-inverse A (1) of the system matrix A is of the form 1 A (1) = P 1 Q v 11 v 12 According to Corollary 23 the general solution of the system (4) is of the form y 1 x = P P 1 y 2, v 11 v 12 1 y 3 where P 1 = Therefore, we obtain x = P v 11 v 12 1 1 2 3 3 6 1 y 1 +2y 2 +3y 3 3y 2 6y 3 = P v 11 y 1 (2v 11 3v 12 y 2 (3v 11 6v 12 1)y 3 If we take α = v 11 y 1 (2v 11 3v 12 y 2 (3v 11 6v 12 1)y 3 as a parameter we get the general solution 2 1 1 3 x = 1 2 3 = 1 α y 3 α 2α α 5

Corollary 25 The vector x = A (1) c is the general solution of the system (1), if and only if the {1}-inverse A (1) of the system matrix A has the form (3) for arbitrary matrices U and W and c the rows of the matrix Vc a are free parameters, where Qc = c = a Remark 26 Similar result can be found in paper B Malešević and B Radičić 3 Example 27 Consider the non-homogeneous linear system x 1 +2x 2 +3x 3 = 7 4x 1 +5x 2 +6x 3 = 8 (5) According to Corollary 25 the general solution of the system (5) is of the form 1 7 x = P 7 1 Q = P 2 8 v 11 v 12 7v 11 2v 12 If we take α = 7v 11 2v 12 as a parameter we obtain the general solution of the system 2 7 1 1 7 19 +α 3 x = P 2 = 1 3 2 2 = 2 2α 3 3 α 1 α α We are now concerned with the matrix equation where A C m n, X C n k and C C m k AX = C, (6) Lemma 28 The matrix equation (6) has a solution if and only if the last m a rows of the matrix C = QC are zeros, where Q C m m is regular matrix such that (2) holds 6

Proof: If we write X = X 1 X 2 X k and C = C 1 C 2 C k, then we can observe the matrix equation (6) as the system of matrix equations AX 1 = C 1 AX 2 = C 2 AX k = C k Each of the matrix equation AX i = C i, 1 i k, by Lemma 21 has solution if and only if the last m a coordinates of the vector C i = QC i are zeros Thus, the previous system has solution if and only if the last m a rows of the matrix C = QC are zeros, which establishes that the matrix equation (6) has solution if and only if all entries of the last m a rows of the matrix C are zeros Theorem 29 The matrix X = A (1) C +(I A (1) A)Y C n k, Y C n k is an arbitrarymatrix, isthe general solutionof the matrix equation (6) if and only if the {1}-inverse A (1) of the system matrix A has the form (3) for arbitrary matrices U and W and the entries of the matrix V(C a Y a )+Y (n a) are mutually independent free parameters, where QC = C = Y P 1 Y = Y a = Y n a C a Proof: ApplyingtheTheorem22ontheeachsystemAX i = C i, 1 i k, we obtain that C a i X i = P V(C a i Y a i )+Y n a i is the general solution of the system if and only if the rows of the matrix V(C a i Y a i )+Y n a i are n a free parameters Assembling these individual solutions together we get that C a X = P V(C a Y a)+y n a 7 and

is the general solution of the matrix equation (6) if and only if entries of the matrix V(C a Y a )+Y n a are (n a)k mutually independent free parameters From now on we proceed with the study of the non-homogeneous linear system of the form xb = d, (7) where B is an n m matrix over the field C of rank b and d is an 1 m matrix over C Let R C n n and S C m m be regular matrices such that Ib RBS = E b = (8) An {1}-inverse of the matrix B can be represented in the Rohde s form B (1) Ib M = S R (9) N K where M = m ij, N = n ij and K = k ij are arbitrary matrices of corresponding dimensions b (n b), (m b) b and (m b) (n b) with mutually independent entries Lemma 21 The non-homogeneous linear system (7) has a solution if and only if the last m b elements of the row d = ds are zeros, where S C m m is regular matrix such that (8) holds Proof: By transposing the system (7) we obtain system B T x T = d T and by transposing the matrix equation (8) we obtain that S T B T R T = E b According to Lemma 21 the system B T x T = d T has solution if and only if the last m b coordinates of the vector S T d T are zeros, ie if and only if the last m b elements of the row d = ds are zeros Theorem 211 The row x = db (1) +y(i BB (1) ), y C 1 n is an arbitrary row, is the general solution of the system (7), if and only if the {1}-inverse B (1) of the system matrix B has the form (9) for arbitrary matrices N and K and the columns of the matrix (d b y b )M+y n b are free parameters, where ds = d = d b and yr 1 = y = y b y n b 8

Proof: The basic idea of the proof is to transpose the system (7) and to apply the Theorem 22 The {1}-inverse of the matrix B T is equal to a transpose of the {1}-inverse of the matrix B Hence, we have (B T ) (1) = (B (1) ) T = ( S Ib M N K ) T Ib N T R = R T M T K T We can now proceed analogously to the proof of the Theorem 22 to obtain that d T x T = R T b M T (d T b y b T)+yT n b is the general solution of the system B T x T = d T if and only if the rows of the matrix M T (d T b y b T)+yT n b are n b free parameters Therefore, x = d b (d b y b )M +y n b R is the general solution of the system (7) if and only if the columns of the matrix (d b y b )M +y n b are n b free parameters Analogous corollaries hold for the Theorem 211 We now deal with the matrix equation where X C k n, B C n m and D C k m S T XB = D, (1) Lemma 212 The matrix equation (1) has a solution if and only if the last m b columns of the matrix D = DS are zeros, where S C m m is regular matrix such that (8) holds Theorem 213 The matrix X = DB (1) +Y(I BB (1) ) C k n, Y C k n is an arbitrarymatrix, isthe general solutionof the matrix equation (1) if and only if the {1}-inverse B (1) of the system matrix B has the form (9) for arbitrary matrices N and K and the entries of the matrix (D b Y b )M +Y (n b) are mutually independent free parameters, where DS = D = D b and YR 1 = Y = Y b Y n b 9

3 An application In this section we will briefly sketch properties of the general solution of the matrix equation AXB = C, (11) where A C m n, X C n k, B C k l and C C m l If we denote by Y matrix product XB, then the matrix equation (11) becomes AY = C (12) According to the Theorem 29 the general solution of the system (12) can be presented as a product of the matrix P and the matrix which has the first a = rank(a) rows same as the matrix QC and the elements of the last m a rows are (m a)n mutually independent free parameters, P and Q are regular matrices such that QAP = E a Thus, we are now turning on to the system of the form XB = D (13) By the Theorem 213 we conclude that the general solution of the system (13) can be presented as a product of the matrix which has the first b = rank(b) columns equal to the first b columns of the matrix DS and the rest of the columns have mutually independent free parameters as entries, and the matrix R, for regular matrices R and S such that RBS = E b Therefore, the general solution of the system (11) is of the form X = P Gab F H L R, where G ab is a submatrix of the matrix QCS corresponding to the first a rows and the first b columns and the entries of the matrices F, H and L are nk ab free parameters We will illustrate this on the following example Example 31 We consider the matrix equation where A = 1 2 2 4, B = take Y = XB, we obtain the system AXB = C, 1 2 1 1 2 1 1 2 1 AY = C 1 and C = 1 2 1 2 4 2 If we

It is easy to check that the matrix A is of the rank a = 1 and for matrices 1 1 2 Q = and P = the equality QAP = E 2 1 1 a holds Based on the Theorem 29, the equation AY = C can be rewritten in the system form 1 AY 1 = 2 2 AY 2 = 4 1 AY 3 = 2 Combining the Theorem 22 with the equality c 1 c 2 c 3 1 1 2 1 = 2 1 2 4 2 yields Y 1 = P 1 v vz 11 +z }{{ 21 } α 2 = Y 2 = P 2v 2vz 12 +z }{{ 22 } β 1 Y 3 = P v vz 13 +z }{{ 23, } Therefore, the general solu- z11 z for an arbitrary matrix Z = 12 z 13 z 21 z 22 z 23 tion of the system AY = C is 1 2 1 Y = P α β γ From now on, we consider the system γ XB = D 1 2 1 for D = P 1 2 1 α β γ = 1+2α 2+2β 1+2γ α β γ 11

There are regular matrices R = 1 1 1 1 1 and S = 1 2 1 1 1 such that RBS = E b holds Since the rank of the matrix B is b = 1, according to the Lemma 212 all entries of the last two columns of the matrix D = DS are zeros, ie we have γ = α, β = 2α Hence, we get that the matrix D is of the form D = X = 1+2α α 1+2α (1+2α t 11 )m 11 +t }{{ 12 } β1 α (α t 21 )m 11 +t }{{ 22 } for an arbitrary matrix T = system AXB = C is 1+2α β1 β X = 2 β 1 β 2 α γ 1 γ 2 γ 1 γ 2 Applying the Theorem 213, we obtain (1+2α t 11 )m 12 +t 13 }{{} β2 R, (α t 12 )m 12 +t }{{ 23 } γ 1 γ 2 t11 t 12 t 13 Finally, the solution of the t 21 t 22 t 23 Acknowledgment Research is partially supported by the Ministry of Science and Education of the Republic of Serbia, Grant No17432 References 1 A Ben Israel and TNE Greville, Generalized Inverses: Theory and Applications, Springer, New York, 23 2 SL Campbell and CD Meyer, Generalized Inverses of Linear Transformations, SIAM series CLASSICS in Applied Mathematics, Philadelphia, 29 3 B Malešević and B Radičić, Non-reproductive and reproductive solutions of some matrix equations, Proceedings of International Conference Mathematical and Informational Technologies 211, pp 246-251, Vrnjačka Banja, 211 12

4 B Malešević and B Radičić, Reproductive and non-reproductive solutions of the matrix equation AXB = C, arxiv:1184867 5 B Malešević and B Radičić, Some considerations of matrix equations using the concept of reproductivity, Kragujevac J Math, Vol 36 (212) No 1, pp 151 161 6 R Penrose, A generalized inverses for matrices, Math Proc Cambridge Philos Soc, Vol 51 (1955), pp 46 413 7 V Perić, Generalized reciprocals of matrices, (in Serbo-Croatian), Matematika (Zagreb), Vol 11 (1982) No 1, pp 4 57 8 CA Rohde, Contribution to the theory, computation and application of generalized inverses, doctoral dissertation, University of North Carolina at Raleigh, 1964 9 NS Urquhart, The nature of the lack of uniqueness of generalized inverse matrices, SIAM Review, Vol 11 (1969) No 2, pp 268 271 13