Calcolo DOI 10.1007/s10092-014-0136-6
Generalized reflexive and anti-reflexive solutions of AX = B Xifu Liu · Ye Yuan
Received: 5 October 2014 / Accepted: 30 December 2014 © Springer-Verlag Italia 2015
Abstract In this paper, we establish some new conditions for the existence and the representations for the (P, Q) generalized reflexive and anti-reflexive solutions to matrix equation AX = B with respect to the generalized reflection matrix dual (P, Q). Moreover, in corresponding solution sets of the equation, the explicit expressions of the nearest matrix to a given matrix in the Frobenius norm have been provided. Keywords (P, Q) generalized reflexive solution · (P, Q) generalized anti-reflexive solution · Matrix equation · Matrix nearness problem Mathematics Subject Classification
15A24
1 Introduction Let Cm×n denote the set of all m × n matrices. For A ∈ Cn×n , its trace will be denoted Frobenius by tr (A). For A ∈ Cm×n , its conjugate transpose, √ √ norm and Moore–Penrose inverse will be denoted by A∗ , A F = tr (A A∗ ) = tr (A∗ A) and A† respectively. In represents the identity matrix of size n. For convenience, we denote E A = I − A A† and FA = I − A† A.
X. Liu (B) School of Science, East China Jiaotong University, Nanchang 330013, China e-mail:
[email protected] Y. Yuan PLA Air Force Troop 94211, Zhengzhou 450000, China e-mail:
[email protected]
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A matrix P ∈ Cn×n is called a generalized reflection matrix if P ∗ = P and P 2 = I . Chen [1] and Chen and Sameh [2] defined two subspaces of matrix: Crm×n (P, Q) = A ∈ Cm×n : A = P AQ , Cam×n (P, Q) = A ∈ Cm×n : A = −P AQ , where P, Q are generalize reflection matrices. The matrices A ∈ Crm×n (P, Q), B ∈ Cam×n (P, Q) are said to be a (P, Q) generalized reflexive and (P, Q) generalized antireflexive matrices respectively with respect to the generalized reflection matrix dual (P, Q). The (P, Q) generalized reflexive and anti-reflexive matrices have applications in system and control theory, in engineering, in scientific computations and various other fields (see [1,2]). Let A ∈ Cm×n , B ∈ Cm×k be known, and X ∈ Cn×k be variable, consider the following equation AX = B. (1.1) It is well known that (1.1) is consistent if and only if A A† B = B, in this case, a general solution can be given by X = A† B + FA V where V ∈ Cn×n is arbitrary. Khatri and Mitra [3] investigated the Hermitian solution and nonnegative definite solution to (1.1). The Re-nonnegative definite (Re-nnd) and Re-positive definite (Repd) solutions were also investigated in [4–6]. For the (P, Q) generalized reflexive and anti-reflexive solutions to (1.1) when n = k, Zhang et al. [7] considered the following two problems by using the structure properties of matrices: the first problem is to consider the (P, Q) generalized reflexive and anti-reflexive solutions with respect to the generalized reflection matrix dual (P, Q); the other is to consider the matrix nearness problem (1.2) min X − C F , X ∈S X
where C ∈ Cn×k is a given matrix, and S X is the solution set of Eq. (1.1). Specially, when P = Q, Peng and Hu [8] studied these two problems. Using the same method, Cvetkovi´c-Ili´c [9] and Peng et al. [10] considered the reflexive and anti-reflexive solutions to AX B = C, Zhou and Yang [11] studied the Hermitian reflexive solutions and the anti-Hermitian reflexive solutions to matrix equations (AX = B, XC = D). Motivated by the above work, in this paper, we restudy these two problem investigated by Zhang et al. [7]. By using the Moore–Penrose inverse, we present sufficient and necessary conditions for the existence and the expressions of the (P, Q) generalized reflexive and anti-reflexive solutions to AX = B. The matrix nearness problem (1.2) is also considered. Before giving the main results, we first introduce several lemmas as follows. The following two results can be easy to verify by the definitions. Lemma 1.1 Let A ∈ Cm×n , B ∈ Ck×n . If B A∗ = 0, then
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A B
†
= A† B † .
Generalized reflexive and anti-reflexive solutions
Lemma 1.2 Let A, B ∈ Cm×n . If B A∗ = 0(or B ∗ A = 0), then A + B2F = A2F + B2F . Lemma 1.3 Let P ∈ Cm×m and Q ∈ Cn×n be two generalized reflection matrices, and A ∈ Cm×n be variable. Then every m × n (P, Q) generalized reflexive matrix can be written as A + P AQ; and every m × n (P, Q) generalized anti-reflexive matrix can be written as A − P AQ.
Proof The proof is trivial. 2 The solution of the matrix Eq. (1.1)
In this section, our purpose is to establish some new conditions for the existence and representations for the (P, Q) generalized reflexive and anti-reflexive solutions to AX = B. For convenience, the following notations will be used in this paper. For A ∈ Cm×n , P ∈ Cn×n is a generalized reflection matrix, we set A1 (P) =
A(I + P) A(I − P) , A2 (P) = , A(I − P) A(I + P)
(2.1)
and denote [Ai (P)]∗ and [Ai (P)]† by Ai∗ (P) and Ai† (P) (i = 1, 2) for short respectively. Next, we will give some properties on A1 (P) and A2 (P) defined by (2.1). Lemma 2.1 Let P ∈ Cn×n be a generalized reflection matrix, and A ∈ Cm×n . Then (1) FA1 (P) ∈ Crn×n (P, P), and AFA1 (P) = 0; (2) FA2 (P) ∈ Crn×n (P, P), and AFA2 (P) = 0. (3) FA1 (P) = FA2 (P) . Proof In view of Lemma 1.1, we have A†1 (P) = [A(I + P)]† [A(I − P)]† , FA1 (P) = In − [A(I + P)]† A(I + P) − [A(I − P)]† A(I − P). Hence, P FA1 (P) P = In − P[A(I + P)]† A(I + P)P − P[A(I − P)]† A(I − P)P = In − [A(I + P)]† A(I + P) − [A(P − I )]† A(P − I ) = In − [A(I + P)]† A(I + P) − [A(I − P)]† A(I − P) = FA1 (P) ,
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means that FA1 (P) is a (P, P) generalized reflexive matrix with respect to the generalized reflection matrix P. Moreover, AFA1 (P) = A − A[A(I + P)]† A(I + P) − A[A(I − P)]† A(I − P) 1 = A − [A(I + P) + A(I − P)][A(I + P)]† A(I + P) 2 1 − [A(I + P) + A(I − P)][A(I − P)]† A(I − P) 2 1 1 = A − A(I + P) − A(I − P) 2 2 = 0. Similarly, the statements (2) and (3) can be deduced.
In order to discuss the (P, Q) generalized reflexive and anti-reflexive solutions to AX = B, we first consider a special case B = 0. Theorem 2.1 Let A ∈ Cm×n be given. Then (1) The general solution X ∈ Crn×k (P, Q) to matrix equation AX = 0 can be expressed as (2.2) X = FA1 (P) V, where V ∈ Crn×k (P, Q) is arbitrary. (2) The general solution X ∈ Can×k (P, Q) to matrix equation AX = 0 can be expressed as X = FA1 (P) W, where W ∈ Can×k (P, Q) is arbitrary. Proof (1) It follows from Lemma 2.1 that X = FA1 (P) V is a (P, Q) generalized reflexive solution to AX = 0 with V ∈ Crn×k (P, Q). On the other hand, suppose Y is an arbitrary (P, Q) generalized reflexive solution to AX = 0, i.e., AY = 0 and PY Q = Y , which implies A1 (P)Y = 0. Therefore, Y = FA1 (P) Y , which is of the form (2.2). That is to say, each (P, Q) generalized reflexive solution to AX = 0 can be formed by (2.2). Similarly, statement (2) can be verified. Theorem 2.2 Let A ∈ Cm×n and B ∈ Cm×k be given. Then (1) AX = B has a solution X ∈ Crn×k (P, Q) if and only if A1 (P)A†1 (P)B1 (Q) = B1 (Q). In this case, a general solution X can be written as X = A†1 (P)B1 (Q) + FA1 (P) V,
(2.3)
where V ∈ Crn×k (P, Q) is arbitrary. (2) AX = B has a solution X ∈ Can×k (P, Q) if and only if A1 (P)A†1 (P)B2 (Q) = B2 (Q). In this case, a general solution X can be written as X = A†1 (P)B2 (Q) + FA1 (P) W,
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where W ∈ Can×k (P, Q) is arbitrary. Proof (1) IF: If A1 (P)A†1 (P)B1 (Q) = B1 (Q), then
A(I + P)A†1 (P)B1 (Q) A(I − P)A†1 (P)B1 (Q)
A A†1 (P)B1 (Q) + A P A†1 (P)B1 (Q) = A A†1 (P)B1 (Q) − A P A†1 (P)B1 (Q) B(I + Q) B + BQ = = , B(I − q) B − BQ
(2.4)
gives that A A†1 (P)B1 (Q) = B, i.e., A†1 (P)B1 (Q) is a solution to AX = B. By computation, we have P A†1 (P)B1 (Q)Q = P [A(I + P)]† [A(I − P)]†
B(I + Q) B(I − Q)
Q
= P[A(I + P)]† B(I + Q)Q + P[A(I − P)]† B(I − Q)Q = [A(I + P)]† B(I + Q) + [A(I − P)]† B(I − Q) = A†1 (P)B1 (Q). So, A†1 (P)B1 (Q) is a (P, Q) generalized reflexive solution to AX = B. For any (P, Q) generalized reflexive solution X to AX = B, we have A[X − A†1 (P)B1 (Q)] = 0. Then, (2.3) is followed by Theorem 2.1-(1). ONLY IF: If (2.3) is a (P, Q) generalized reflexive solution to AX = B, then A A†1 (P)B1 (Q) = B, and A P A†1 (P)B1 (Q) = B Q. According to (2.4), A1 (P)A†1 (P)B1 (Q) = B1 (Q) is evident. The proof of statement (2) is similar to (1), hence, the details are omitted.
3 The solution of the matrix nearness problem (1.2) In this section, we deduce the explicit expressions of the solution of the matrix nearness problem (1.2). First, we verify the following results. Lemma 3.1 Let C ∈ Cn×k , FA1 (P) ∈ Crn×n (P, P), V ∈ Crn×k (P, Q) and W ∈ Can×k (P, Q). Then
FA
2
2 V − FA1 (P) C F = FA1 (P) V − FA1 (P) PC Q F (3.1)
2
1 1 ∗ ∗
=
FA1 (P) V − 2 FA1 (P) (C + PC Q) + 2 tr FA1 (P) (CC − C QC P)FA1 (P) . F (3.2) 1 (P)
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Similarly,
FA
2
2 W − FA1 (P) C F = FA1 (P) W + FA1 (P) PC Q F (3.3)
2
1 1 ∗ ∗
=
FA1 (P) W − 2 FA1 (P) (C − PC Q) + 2 tr FA1 (P) (CC + C QC P)FA1 (P) . F (3.4) 1 (P)
Proof Since · F is an unitarily invariant norm, according to the assumptions, (3.1) is obvious. Moreover, we have tr FA1 (P) V QC ∗ P FA1 (P) = tr P FA1 (P) V QC ∗ P FA1 (P) P = tr FA1 (P) P V QC ∗ FA1 (P) = tr FA1 (P) V C ∗ FA1 (P) , tr FA1 (P) PC QV ∗ FA1 (P) = tr P FA1 (P) PC QV ∗ FA1 (P) P = tr FA1 (P) C V ∗ FA1 (P) , tr FA1 (P) C QC ∗ P FA1 (P) = tr P FA1 (P) C QC ∗ P FA1 (P) P = tr FA1 (P) PC QC ∗ FA1 (P) , tr FA1 (P) PCC ∗ P FA1 (P) = tr FA1 (P) CC ∗ FA1 (P) . Therefore,
2
FA (P) V − 1 FA (P) (C + PC Q)
1
1 2 F
∗ 1 1 = tr FA1 (P) V − FA1 (P) (C + PC Q) FA1 (P) V − FA1 (P) (C + PC Q) 2 2
1 1 = tr FA1 (P) V V ∗ FA1 (P) − FA1 (P) V C ∗ FA1 (P) − FA1 (P) V QC ∗ P FA1 (P) 2 2 1 1 1 − FA1 (P) C V ∗ FA1 (P) − FA1 (P) PC QV ∗ FA1 (P) + FA1 (P) CC ∗ FA1 (P) 2 2 4 1 1 1 ∗ ∗ ∗ + FA1 (P) C QC P FA1 (P) + FA1 (P) PC QC FA1 (P) + FA1 (P) PCC P FA1 (P) 4 4 4 ∗ ∗ = tr FA1 (P) V V FA1 (P) − FA1 (P) V C FA1 (P) − FA1 (P) C V ∗ FA1 (P) 1 1 ∗ ∗ + FA1 (P) CC FA1 (P) + FA1 (P) C PC P FA1 (P) 2 2
2 1 = FA1 (P) V − FA1 (P) C F − tr FA1 (P) (CC ∗ − C QC ∗ P)FA1 (P) . 2 Hence, (3.2) is evident. Equation (3.3) and (3.4) can be verified similarly.
The following theorem give the explicit expressions of the solutions of the matrix nearness problem (1.2).
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Theorem 3.1 Given a matrix C ∈ Cn×k . (1) Assume the solution set S X ⊆ Crn×k (P, Q) of Eq. (1.1) is nonempty, then the matrix nearness problem (1.2) has an unique solution Xˆ in S X , which can be written as 1 (3.5) Xˆ = A†1 (P)B1 (Q) + FA1 (P) (C + PC Q). 2 (2) Assume the solution set S X ⊆ Can×k (P, Q) of Eq. (1.1) is nonempty, then the matrix nearness problem (1.2) has an unique solution Xˆ in S X , which can be written as 1 Xˆ = A†1 (P)B2 (Q) + FA1 (P) (C − PC Q). 2 Proof (1) Let X ∈ S X , it follows from Lemma 1.2 and (2.3) that
2
X − C2F = A†1 (P)B1 (Q) + FA1 (P) V − C
F
2
2
†
† = A1 (P)B1 (Q) − A1 (P)A1 (P)C + FA1 (P) V − FA1 (P) C F F
2
†
† = A1 (P)B1 (Q)− A1 (P)A1 (P)C
F
1 + tr FA1 (P) (CC ∗ −C QC ∗ P)FA1 (P) 2
2
1
F +
F V − (C + PC Q) A1 (P)
A1 (P)
2 F
Therefore, there exists Xˆ ∈ S X such that the matrix nearness problem (1.2) holds if and only if there exists V ∈ Crn×k (P, Q) such that
1
min
FA1 (P) V − 2 FA1 (P) (C + PC Q) . n×k V ∈Cr (P,Q) F Obviously, matrix equation FA1 (P) V = 21 FA1 (P) (C + PC Q) is consistent for V ∈ Crn×k (P, Q), for example, take V = 21 (C + PC Q). Hence,
FA (P) V − 1 FA (P) (C + PC Q) = 0. 1 1
2 V ∈Crn×k (P,Q) F min
Therefore, (3.5) is evident. Similarly, statement (2) can be obtained.
Acknowledgments The authors would like to thank the Editor and the referees for their very detailed comments and valuable suggestions which greatly improved our presentation. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11426107, 11461026, 51465018, 61472138), and the S&T plan projects of Jiangxi Provincial Department of Education (Grant No. GJJ14380).
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