Reproductive and non-reproductive solutions of the matrix equation AXB=C
aa r X i v : . [ m a t h . R A ] A ug Reproductive and non-reproductive solutionsof the matrix equation
AX B = C Branko Maleˇsevi´c ∗ and Biljana Radiˇci´c ∗ University of Belgrade, Faculty of Electrical Engineering,Department of Applied Mathematics, Serbia
Abstract.
In this article we consider a consistent matrix equation
AXB = C whena particular solution X is given and we represent a new form of the general solutionwhich contains both reproductive and non-reproductive solutions (it depends on theform of particular solution X ). We also analyse the solutions of some matrix systemsusing the concept of reproductivity and we give a new form of the condition for theconsistency of the matrix equation AXB = C . Keywords: Reproductive equations, reproductive solutions, matrix equation
AXB = C The concept of the reproductive equations was introduced by
S. B. Preˇsi´c [4].
Definition 1.2.
The reproductive equations are the equations of the follow-ing form: x = f ( x ) , (1)where x is a unknown, S is a given set and f : S −→ S is a given function whichsatisfies the following condition: f ◦ f = f. (2)The condition (2) is called the condition of reproductivity . The most im-portant statements in relation to the reproductive equations are given by thefollowing two theorems [4] (see also [5], [6] and [12]): Theorem 1.1. ( S. B. Preˇsi´c ) For any consistent equation J ( x ) there is anequation of the form x = f ( x ), which is equivalent to J ( x ) being in the sametime reproductive as well. (cid:7) Theorem 1.2. ( S. B. Preˇsi´c ) If a certain equation J ( x ) is equivalent tothe reproductive one x = f ( x ), the general solution is given by the formula x = f ( y ), for any value y ∈ S . (cid:7) The concept of the reproductive equations allows us to analyse the solutionsof some matrix systems (see Application 2.1. and Application 2.2. in the fol-lowing section of this paper). In [7], [8] and [11] authors considered the generalapplications of the concept of reproductivity.1
The matrix equation
AX B = C Let m, n ∈ N and C is the field of complex numbers. The set of all matrices oforder m × n over C is denoted by C m × n . For the set of all matrices from C m × n with a rank a we use denotement C m × na . Let A = [ a i,j ] ∈ C m × n . By A i → wedenote the i -th row of A , i = 1 , ..., m . For the j -th column of A , j = 1 , ..., n , weuse denotement A ↓ j .A solution of the matrix equation AXA = A (3)is called { } -inverse of the matrix A and it is denoted by A (1) . The set of all { } -inverses of the matrix A is denoted by A { } . For the matrix A , let regularmatrices Q ∈ C m × m and P ∈ C n × n be determined so that the following equalityis true. QAP = E a = (cid:20) I a
00 0 (cid:21) , (4)where a = rank ( A ). In [3] C. Rohde showed that the general { } -inverse A (1) can be represented in the following form: A (1) = P (cid:20) I a X X X (cid:21) Q, (5)where X , X and X are arbitrary matrices of suitable sizes.Let A ∈ C m × n , B ∈ C p × q and C ∈ C m × q . In the paper [1] R. Penrose proved the following theorem related to the matrix equation
AXB = C. (6) Theorem 2.1. ( R. Penrose ) The matrix equation (6) is consistent iff forsome choice of { } -inverses A (1) and B (1) of the matrices A and B the condition AA (1) CB (1) B = C (7)is true. The general solution of the matrix equation (6) is given by the formula X = f ( Y ) = A (1) CB (1) + Y − A (1) AY BB (1) , (8)where Y is an arbitrary matrix of suitable size. (cid:7) If a particular solution X of the matrix equation (6) is given, the formula ofgeneral solution is given in the following theorem. Theorem 2.2. If X is a particular solution of the matrix equation (6),then the general solution of the matrix equation (6) is given by the formula X = g ( Y ) = X + Y − A (1) AY BB (1) , (9)where Y is an arbitrary matrix of suitable size. The function g satisfies thecondition of reproductivity (2) iff X = A (1) CB (1) . Proof.
See [14] and [16] (where different proofs are given). (cid:7) X . Remark 2.1.
In the paper [14] authors proved that there is a matrix equa-tion (6) and a particular solution X so that: X = A (1) CB (1) , (10)for any choice of { } -inverses A (1) and B (1) . In that case the formula (9) givesthe general non-reproductive solution. Otherwise, the formula (9) gives thegeneral reproductive solution. (cid:7) Example 2.1.
Compared to [14], we give a simpler example of the matrixequation (6) and a particular solution X so that (10) is valid. Let’s considerthe matrix equation: (cid:2) (cid:3) X (cid:20) (cid:21) = [12] , (11)with A = (cid:2) (cid:3) , B = (cid:2) (cid:3) T , C = (cid:2) (cid:3) . Then, there is a particular solution: X = (cid:20) − −
36 12 (cid:21) (12)of the previous matrix equation. It is easy to show that A (1) = (cid:2) − a a (cid:3) T ( a ∈ C ) , B (1) = (cid:2) − b b (cid:3) ( b ∈ C ) . So, X = A (1) CB (1) = (cid:20) − a − b +72 ab b − ab a − ab ab (cid:21) . (13)From AXB − C = 0 ⇐⇒ x , + 3 x , + 2 x , + 6 x , −
12 = 0, where X = (cid:2) x i,j (cid:3) ,we get that the matrix of general solution is given by the following form: X = (cid:20) − p − q − r pq r (cid:21) , ( p, q, r ∈ C ) . (14)For p = − , q = −
36 and r = 12, we get the particular solution X of thematrix equation (11), but X = X = A (1) CB (1) for any choice of { }− inverses A (1) and B (1) , because from X = X we obtain the contradiction ( ab = 1 and a = 0 , b = 0). (cid:7) In [2]
S. B. Preˇsi´c analysed the matrix equation (3) and he proved thefollowing theorem ♮ ) . Theorem 2.3.
For any square matrix A ∈ C n × n and any general { } -inverse A (1) the following equivalences are true: ( E ) AX = 0 ⇐⇒ ( ∃ Y ∈ C n × n ) X = Y − A (1) AY , ( E ) XA = 0 ⇐⇒ ( ∃ Y ∈ C n × n ) X = Y − Y AA (1) , ( E ) AXA = A ⇐⇒ ( ∃ Y ∈ C n × n ) X = A (1) + Y − A (1) AY AA (1) , ( E ) AX = A ⇐⇒ ( ∃ Y ∈ C n × n ) X = I + Y − A (1) AY , ( E ) XA = A ⇐⇒ ( ∃ Y ∈ C n × n ) X = I + Y − Y AA (1) . (cid:7) In the general case the general solutions ( E )-( E ) of Theorem 2.3. do notdirectly appear according to Penrose ’s theorem. In [9]
M. Haveri´c showedthat we can get
Penrose ’s solutions from
Preˇsi´c ’s solutions. She proved thefollowing statement. ♮ ) with the first appearances of non-reproductive solutions (cid:0) see ( E )-( E ) (cid:1) heorem 2.4. For any square matrix A ∈ C n × n and any general { }− inverse A (1) the following equivalences are true. ( E ) AX = 0 ⇐⇒ ( ∃ Y ∈ C n × n ) X = Y − A (1) AY , ( E ) XA = 0 ⇐⇒ ( ∃ Y ∈ C n × n ) X = Y − Y AA (1) , ( E ′ ) AXA = A ⇐⇒ ( ∃ Y ∈ C n × n ) X = A (1) AA (1) + Y − A (1) AY AA (1) , ( E ′ ) AX = A ⇐⇒ ( ∃ Y ∈ C n × n ) X = A (1) A + Y − A (1) AY , ( E ′ ) XA = A ⇐⇒ ( ∃ Y ∈ C n × n ) X = AA (1) + Y − Y AA (1) . (cid:7) Let’s note that the previous two theorems are special case of Theorem 2.2.For a consistent matrix equation (6) the following equivalence is true:
AXB = C ⇐⇒ X = f ( X ) = X − A (1) ( AXB − C ) B (1) . (15)Therefore, based on Theorem 1.2., we have a short proof of the generality offormula (7) in Theorem 2.1. (see [16]).In the following applications we analysed the solutions of some matrix sys-tems using the concept of reproductivity. Application 2.1.
Let
A, B, D and E be given complex matrices of suitablesizes. If the following matrix system is consistent: AX = B ∧ XD = E, (16)then the general solution is given by the formula ([13], A. Ben-Israel and
T. N. E. Greville ) X = g ( Y ) = X + ( I − A (1) A ) Y ( I − DD (1) ) , (17)where Y is an arbitrary matrix of suitable size.In [16] authors proved that if the matrix system (16) is consistent, the generalreproductive solution is given by the formula X = f ( Y ) = A (1) B + ED (1) − A (1) AED (1) + ( I − A (1) A ) Y ( I − DD (1) ) , (18)where Y is an arbitrary matrix of suitable size. The proof is based on theequivalence ( AX = B ∧ XD = E ) ⇐⇒ X = f ( X ) (19)and Theorem 1.2. A more detailed proof can be found in [16]. (cid:7) Application 2.2.
Let A ∈ C n × n be a singular matrix. If the followingmatrix system is consistent: AXA = A ∧ AX = XA, (20)then the general solution is given by the formula ([10],
J. D. Keˇcki´c ) X = f ( Y ) = Y + ¯ AA ¯ A − ¯ AAY − Y A ¯ A + ¯ AAY A ¯ A, (21)where Y is an arbitrary matrix of suitable size and ¯ A is a commutative { } -inverse.In [16] authors proved that (21) represents the general solution of (20) usingthe concept of reproductivity. Further applications of the concept of reproduc-tivity for some matrix equations and systems is considered in paper [15]. (cid:7) b A = T A A, b B = BT B and b C = T A CT B (22)where T A is a permutation matrix which permutes linearly independent rowsof the matrix A at the first a positions and T B is a permutation matrix whichpermutes linearly independent columns of the matrix B at the first b positions.Therefore, the matrix b A has linearly independent rows at the first a positionsand the matrix b B has linearly independent columns at the first b positions.Let b A i → = a X l =1 α i,l b A l → , i = a + 1 , ..., m (23)and b B ↓ j = b X k =1 β k,j b B ↓ k , j = b + 1 , ..., q. (24)for some scalars α i,l and β k,j . Then, the following theorem is true. Theorem 2.5.
Let A ∈ C m × na , B ∈ C p × qb and C ∈ C m × q . Suppose that b A , b B and b C are determined by (22) and that (23) and (24) are satisfied. Then, thecondition (7) is true for any choice of { } -inverses A (1) and B (1) iff b C = c , ... c ,b b X k =1 β k,b +1 c ,k ... b X k =1 β k,q c ,k . ... . . ... .. ... . . ... .. ... . . ... .c a, ... c a,b b X k =1 β k,b +1 c a,k ... b X k =1 β k,q c a,k a X l =1 α a +1 ,l c l, ... a X l =1 α a +1 ,l c l,b a X l =1 b X k =1 α a +1 ,l β k,b +1 c l,k ... a X l =1 b X k =1 α a +1 ,l β k,q c l,k . ... . . ... .. ... . . ... .. ... . . ... . a X l =1 α m,l c l, ... b X k =1 α m,l c l,b a X l =1 b X k =1 α m,l β k,b +1 c l,k ... a X l =1 b X k =1 α m,l β k,q c l,k ,where c i,j are arbitrary elements of C . Proof.
The proof can be found in [16]. (cid:7)
Acknowledgements.
The research is partially supported by the Ministry ofEducation and Science, Serbia, Grant No. 174032.
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