Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 593549, 11 pages http://dx.doi.org/10.1155/2013/593549
Research Article Solving Optimization Problems on Hermitian Matrix Functions with Applications Xiang Zhang and Shu-Wen Xiang Department of Computer Science and Information, Guizhou University, Guiyang 550025, China Correspondence should be addressed to Xiang Zhang;
[email protected] Received 17 October 2012; Accepted 20 March 2013 Academic Editor: K. Sivakumar Copyright © 2013 X. Zhang and S.-W. Xiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the extremal inertias and ranks of the matrix expressions 𝑓(𝑋, 𝑌) = 𝐴 3 − 𝐵3 𝑋 − (𝐵3 𝑋)∗ − 𝐶3 𝑌𝐷3 − (𝐶3 𝑌𝐷3 )∗ , where 𝐴 3 = 𝐴∗3 , 𝐵3 , 𝐶3 , and 𝐷3 are known matrices and 𝑌 and 𝑋 are the solutions to the matrix equations 𝐴 1 𝑌 = 𝐶1 , 𝑌𝐵1 = 𝐷1 , and 𝐴 2 𝑋 = 𝐶2 , respectively. As applications, we present necessary and sufficient condition for the previous matrix function 𝑓(𝑋, 𝑌) to be positive (negative), non-negative (positive) definite or nonsingular. We also characterize the relations between the Hermitian part of the solutions of the above-mentioned matrix equations. Furthermore, we establish necessary and sufficient conditions for the solvability of the system of matrix equations 𝐴 1 𝑌 = 𝐶1 , 𝑌𝐵1 = 𝐷1 , 𝐴 2 𝑋 = 𝐶2 , and 𝐵3 𝑋 + (𝐵3 𝑋)∗ + 𝐶3 𝑌𝐷3 + (𝐶3 𝑌𝐷3 )∗ = 𝐴 3 , and give an expression of the general solution to the above-mentioned system when it is solvable.
1. Introduction Throughout, we denote the field of complex numbers by C, the set of all 𝑚 × 𝑛 matrices over C by C𝑚×𝑛 , and the set of . The symbols 𝐴∗ and all 𝑚 × 𝑚 Hermitian matrices by C𝑚×𝑚 ℎ R(𝐴) stand for the conjugate transpose, the column space of a complex matrix 𝐴 respectively. 𝐼𝑛 denotes the 𝑛 × 𝑛 identity matrix. The Moore-Penrose inverse [1] 𝐴† of 𝐴, is the unique solution 𝑋 to the four matrix equations: (i) 𝐴𝑋𝐴 = 𝐴, (ii) 𝑋𝐴𝑋 = 𝑋, (iii) (𝐴𝑋)∗ = 𝐴𝑋,
symbols 𝑖+ (𝐴) and 𝑖− (𝐴) are called the positive index and the negative index of inertia, respectively. For two Hermitian matrices 𝐴 and 𝐵 of the same sizes, we say 𝐴 ≥ 𝐵 (𝐴 ≤ 𝐵) in the L¨owner partial ordering if 𝐴 − 𝐵 is positive (negative) semidefinite. The Hermitian part of 𝑋 is defined as 𝐻(𝑋) = 𝑋 + 𝑋∗ . We will say that 𝑋 is Re-nnd (Re-nonnegative semidefinite) if 𝐻(𝑋) ≥ 0, 𝑋 is Re-pd (Re-positive definite) if 𝐻(𝑋) > 0, and 𝑋 is Re-ns if 𝐻(𝑋) is nonsingular. It is well known that investigation on the solvability conditions and the general solution to linear matrix equations is very active (e.g., [2–9]). In 1999, Braden [10] gave the general solution to
(1) 𝐵𝑋 + (𝐵𝑋)∗ = 𝐴.
(3)
∗
(iv) (𝑋𝐴) = 𝑋𝐴. Moreover, 𝐿 𝐴 and 𝑅𝐴 stand for the projectors 𝐿 𝐴 = 𝐼 − 𝐴† 𝐴, 𝑅𝐴 = 𝐼 − 𝐴𝐴† induced by 𝐴. It is well known that the eigenvalues of a Hermitian matrix 𝐴 ∈ C𝑛×𝑛 are real, and the inertia of 𝐴 is defined to be the triplet I𝑛 (𝐴) = {𝑖+ (𝐴) , 𝑖− (𝐴) , 𝑖0 (𝐴)} ,
(2)
where 𝑖+ (𝐴), 𝑖− (𝐴), and 𝑖0 (𝐴) stand for the numbers of positive, negative, and zero eigenvalues of 𝐴, respectively. The
In 2007, Djordjevi´c [11] considered the explicit solution to (3) for linear bounded operators on Hilbert spaces. Moreover, Cao [12] investigated the general explicit solution to 𝐵𝑋𝐶 + (𝐵𝑋𝐶)∗ = 𝐴.
(4)
Xu et al. [13] obtained the general expression of the solution of operator equation (4). In 2012, Wang and He [14] studied
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some necessary and sufficient conditions for the consistence of the matrix equation ∗
∗
𝐴 1 𝑋 + (𝐴 1 𝑋) + 𝐵1 𝑌𝐶1 + (𝐵1 𝑌𝐶1 ) = 𝐸1
(5)
and presented an expression of the general solution to (5). Note that (5) is a special case of the following system: 𝐴 1 𝑌 = 𝐶1 ,
𝑌𝐵1 = 𝐷1 ,
𝐴 2 𝑋 = 𝐶2 ,
∗
(6)
∗
𝐵3 𝑋 + (𝐵3 𝑋) + 𝐶3 𝑌𝐷3 + (𝐶3 𝑌𝐷3 ) = 𝐴 3 .
To our knowledge, there has been little information about (6). One goal of this paper is to give some necessary and sufficient conditions for the solvability of the system of matrix (6) and present an expression of the general solution to system (6) when it is solvable. In order to find necessary and sufficient conditions for the solvability of the system of matrix equations (6), we need to consider the extremal ranks and inertias of (10) subject to (13) and (11). There have been many papers to discuss the extremal ranks and inertias of the following Hermitian expressions: ∗
2. Extremal Ranks and Inertias of Hermitian Matrix Function (10) with Some Restrictions In this section, we consider formulas for the extremal ranks and inertias of (10) subject to (11) and (13). We begin with the following Lemmas. Lemma 1 (see [21]). (a) Let 𝐴 1 , 𝐶1 , 𝐵1 , and 𝐷1 be given. Then the following statements are equivalent: (1) system (13) is consistent, (2) 𝑅𝐴 1 𝐶1 = 0,
𝐷1 𝐿 𝐵1 = 0,
𝐴 1 𝐷1 = 𝐶1 𝐵1 .
(3) 𝑟 [𝐴 1 𝐶1 ] = 𝑟 (𝐴 1 ) , [
𝐷1 ] = 𝑟 (𝐵1 ) , 𝐵1
(7)
𝑔 (𝑌) = 𝐴 − 𝐵𝑌𝐶 − (𝐵𝑌𝐶)∗ ,
(8)
In this case, the general solution can be written as
ℎ (𝑋, 𝑌) = 𝐴 1 − 𝐵1 𝑋𝐵1∗ − 𝐶1 𝑌𝐶1∗ ,
(9)
𝑌 = 𝐴†1 𝐶1 + 𝐿 𝐴 1 𝐷1 𝐵1† + 𝐿 𝐴 1 𝑉𝑅𝐵1 ,
∗
∗
Tian has contributed much in this field. One of his works [15] considered the extremal ranks and inertias of (7). He and Wang [16] derived the extremal ranks and inertias of (7) subject to 𝐴 1 𝑋 = 𝐶1 , 𝐴 2 𝑋𝐵2 = 𝐶2 . Liu and Tian [17] studied the extremal ranks and inertias of (8). Chu et al. [18] and Liu and Tian [19] derived the extremal ranks and inertias of (9). Zhang et al. [20] presented the extremal ranks and inertias of (9), where 𝑋 and 𝑌 are Hermitian solutions of 𝐴 2 𝑋 = 𝐶2 ,
(11)
𝑌𝐵2 = 𝐷2 ,
(12)
respectively. He and Wang [16] derived the extremal ranks and inertias of (10). We consider the extremal ranks and inertias of (10) subject to (11) and 𝐴 1 𝑌 = 𝐶1 ,
𝑌𝐵1 = 𝐷1 ,
(13)
which is not only the generalization of the above matrix functions, but also can be used to investigate the solvability conditions for the existence of the general solution to the system (6). Moreover, it can be applied to characterize the relations between Hermitian part of the solutions of (11) and (13). The remainder of this paper is organized as follows. In Section 2, we consider the extremal ranks and inertias of (10) subject to (11) and (13). In Section 3, we characterize the relations between the Hermitian part of the solution to (11) and (13). In Section 4, we establish the solvability conditions for the existence of a solution to (6) and obtain an expression of the general solution to (6).
(15)
𝐴 1 𝐷1 = 𝐶1 𝐵1 .
𝑝 (𝑋) = 𝐴 3 − 𝐵3 𝑋 − (𝐵3 𝑋) ,
𝑓 (𝑋, 𝑌) = 𝐴 3 − 𝐵3 𝑋 − (𝐵3 𝑋) − 𝐶3 𝑌𝐷3 − (𝐶3 𝑌𝐷3 ) . (10)
(14)
(16)
where 𝑉 is arbitrary. (b) Let 𝐴 2 and 𝐶2 be given. Then the following statements are equivalent: (1) equation (11) is consistent, (2) 𝑅𝐴 2 𝐶2 = 0,
(17)
𝑟 [𝐴 2 𝐶2 ] = 𝑟 (𝐴 2 ) .
(18)
(3)
In this case, the general solution can be written as 𝑋 = 𝐴† 𝐶 + 𝐿 𝐴𝑊,
(19)
where 𝑊 is arbitrary. Lemma 2 ([22, Lemma 1.5, Theorem 2.3]). Let 𝐴 ∈ C𝑚×𝑚 , ℎ , and denote that 𝐵 ∈ C𝑚×𝑛 , and 𝐷 ∈ C𝑛×𝑛 ℎ 𝐴 𝐵 𝑀 = [ ∗ ], 𝐵 0 𝑁=[ 𝐿=[ 𝐺=[
0 𝑄 ], 𝑄∗ 0
𝐴 𝐵 ], 𝐵∗ 𝐷
𝑃 𝑀𝐿 𝑁 ]. 𝐿 𝑁𝑀∗ 0
(20)
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3 (h) any 𝑋 ∈ 𝐻 satisfies 𝑋 ≥ 0 (𝑋 ≤ 0) if and only if max𝑋∈𝐻𝑖− (𝑋) = 0 (max𝑋∈𝐻𝑖+ (𝑋) = 0).
Then one has the following (a) the following equalities hold 𝑖± (𝑀) = 𝑟 (𝐵) + 𝑖± (𝑅𝐵 𝐴𝑅𝐵 ) ,
(21)
𝑖± (𝑁) = 𝑟 (𝑄) ,
(22)
Lemma 6 (see [16]). Let 𝑝(𝑋, 𝑌) = 𝐴 − 𝐵𝑋 − (𝐵𝑋)∗ − 𝐶𝑌𝐷 − (𝐶𝑌𝐷)∗ , where 𝐴, 𝐵, 𝐶, and 𝐷 are given with appropriate sizes, and denote that 𝐴 𝐵 𝐶 𝑀1 = [ 𝐵∗ 0 0 ] , ∗ [𝐶 0 0 ]
(b) if R(𝐵) ⊆ R(𝐴), then 𝑖± (𝐿) = 𝑖± (𝐴) + 𝑖± (𝐷 − 𝐵∗ 𝐴† 𝐵)). Thus 𝑖± (𝐿) = 𝑖± (𝐴) if and only if R(𝐵) ⊆ R(𝐴) and 𝑖± (𝐷 − 𝐵∗ 𝐴† 𝐵) = 0,
𝐴 𝐵 𝐷∗ 𝑀2 = [𝐵∗ 0 0 ] , [𝐷 0 0 ]
(c) 𝑃 𝑀 0 𝑖± (𝐺) = [𝑀∗ 0 𝑁∗ ] − 𝑟 (𝑁) . [ 0 𝑁 0 ]
(23)
𝑀3 = [
(a) 𝑟[𝐴 𝐵] = 𝑟(𝐴) + 𝑟(𝐸𝐴 𝐵) = 𝑟(𝐵) + 𝑟(𝐸𝐵 𝐴),
(d) (e)
𝑟 [ 𝐸𝐶𝐵 𝐴 ]
(f)
𝑟 [ 𝐸𝐴𝐸 𝐶 𝐵𝐹0𝐷 ]
(b) (c)
=
𝐶 0] 𝑟[𝐴 𝐵
=
− 𝑟(𝐵),
𝐴 𝐵 0 𝑟[𝐶 0 𝐸] 0 𝐷 0
𝐴 𝐵 𝐶 𝐷∗ [ 𝑀5 = 𝐵∗ 0 0 0 ] . [𝐷 0 0 0 ] Then one has the following: (1) the maximal rank of 𝑝(𝑋, 𝑌) is
− 𝑟(𝐷) − 𝑟(𝐸),
, 𝐵 ∈ C𝑚×𝑛 , 𝐶 ∈ C𝑛×𝑛 Lemma 4 (see [15]). Let 𝐴 ∈ C𝑚×𝑚 ℎ ℎ ,𝑄 ∈ C𝑚×𝑛 , and 𝑃 ∈ C𝑝×𝑛 be given, and 𝑇 ∈ C𝑚×𝑚 be nonsingular. Then one has the following
max 𝑟 [𝑝 (𝑋, 𝑌)] = min {𝑚, 𝑟 (𝑀1 ) , 𝑟 (𝑀2 ) , 𝑟 (𝑀3 )} ,
𝑋∈C𝑛×𝑚 ,𝑌
(25) (2) the minimal rank of 𝑝(𝑋, 𝑌) is
∗
(1) 𝑖± (𝑇𝐴𝑇 ) = 𝑖± (𝐴),
min 𝑟 [𝑝 (𝑋, 𝑌)]
𝑋∈C𝑛×𝑚 ,𝑌
(2) 𝑖± [ 𝐴0 𝐶0 ] = 𝑖± (𝐴) + 𝑖± (𝐶), (3) 𝑖± [ 𝑄0∗ 𝐴
𝑄 0]
(4) 𝑖± [ 𝐿 𝑃 𝐵∗
(26)
= 2𝑟 (𝑀3 ) − 2𝑟 (𝐵)
= 𝑟(𝑄),
𝐵𝐿 𝑃 0 ]
(24)
𝐴 𝐵 𝐶 𝐷∗ [ 𝑀4 = 𝐵∗ 0 0 0 ] , ∗ [𝐶 0 0 0 ]
Lemma 3 (see [23]). Let 𝐴 ∈ C𝑚×𝑛 , 𝐵 ∈ C𝑚×𝑘 , and 𝐶 ∈ C𝑙×𝑛 . Then they satisfy the following rank equalities: 𝑟[𝐴 𝐶 ] = 𝑟(𝐴) + 𝑟(𝐶𝐹𝐴 ) = 𝑟(𝐶) + 𝑟(𝐴𝐹𝐶 ), 𝐴 𝑟 [ 𝐶 𝐵0 ] = 𝑟(𝐵) + 𝑟(𝐶) + 𝑟(𝐸𝐵 𝐴𝐹𝐶), 𝑟[𝐵 𝐴𝐹𝐶] = 𝑟 [ 𝐵0 𝐴 𝐶 ] − 𝑟(𝐶),
𝐴 𝐵 𝐶 𝐷∗ ], 𝐵∗ 0 0 0
𝐴 𝐵 0 0 𝑃∗ 0 𝑃 0
+ 𝑟(𝑃) = 𝑖± [ 𝐵∗
+ max {𝑢+ + 𝑢− , V+ + V− , 𝑢+ + V− , 𝑢− + V+ } ,
].
Lemma 5 (see [22, Lemma 1.4]). Let 𝑆 be a set consisting of (square) matrices over C𝑚×𝑚 , and let 𝐻 be a set consisting of . Then Then one has the following (square) matrices over C𝑚×𝑚 ℎ (a) 𝑆 has a nonsingular matrix if and only if max𝑋∈𝑆 𝑟(𝑋) = 𝑚; (b) any 𝑋 ∈ 𝑆 is nonsingular if and only if min𝑋∈𝑆 𝑟(𝑋) = 𝑚;
(3) the maximal inertia of 𝑝(𝑋, 𝑌) is max 𝑖± [𝑝 (𝑋, 𝑌)] = min {𝑖± (𝑀1 ) , 𝑖± (𝑀2 )} ,
𝑋∈C𝑛×𝑚 ,𝑌
(4) the minimal inertias of 𝑝(𝑋, 𝑌) is min 𝑖± [𝑝 (𝑋, 𝑌)] = 𝑟 (𝑀3 ) − 𝑟 (𝐵)
𝑋∈C𝑛×𝑚 ,𝑌
(c) {0} ∈ 𝑆 if and only if min𝑋∈𝑆 𝑟(𝑋) = 0;
+ max {𝑖± (𝑀1 ) − 𝑟 (𝑀4 ) ,
(d) 𝑆 = {0} if and only if max𝑋∈𝑆 𝑟(𝑋) = 0; (e) 𝐻 has a matrix 𝑋 > 0 (𝑋 < 0) if and only if max𝑋∈𝐻𝑖+ (𝑋) = 𝑚(max𝑋∈𝐻𝑖− (𝑋) = 𝑚); (f) any 𝑋 ∈ 𝐻 satisfies 𝑋 > 0 (𝑋 < 0) if and only if min𝑋∈𝐻𝑖+ (𝑋) = 𝑚 (min𝑋∈𝐻𝑖− (𝑋) = 𝑚); (g) 𝐻 has a matrix 𝑋 ≥ 0 (𝑋 ≤ 0) if and only if min𝑋∈𝐻𝑖− (𝑋) = 0 (min𝑋∈𝐻𝑖+ (𝑋) = 0);
(27)
(28)
𝑖± (𝑀2 ) − 𝑟 (𝑀5 )} , where 𝑢± = 𝑖± (𝑀1 ) − 𝑟 (𝑀4 ) ,
V± = 𝑖± (𝑀2 ) − 𝑟 (𝑀5 ) . (29)
Now we present the main theorem of this section.
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Theorem 7. Let 𝐴 1 ∈ C𝑚×𝑛 ,𝐶1 ∈ C𝑚×𝑘 , 𝐵1 ∈ C𝑘×𝑙 , 𝐷1 ∈ C𝑛×𝑙 , 𝑝×𝑝 𝐴 2 ∈ C𝑡×𝑞 , 𝐶2 ∈ C𝑡×𝑝 ,𝐴 3 ∈ Cℎ , 𝐵3 ∈ C𝑝×𝑞 , 𝐶3 ∈ C𝑝×𝑛 , 𝑝×𝑛 and 𝐷3 ∈ C be given, and suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by 𝑆 and (11) by 𝐺. Put 𝐴3 [ 𝐶∗ [ 3 𝐸1 = [ 3 [𝐶1 𝐷 [ 𝐵3∗ [ 𝐶2
𝐶3 𝐷3∗ 𝐶1∗ 0 𝐴∗1 𝐴1 0 0 0 0 0
𝐵3 𝐶2∗ 0 0] ] 0 0] ], 0 𝐴∗2 ] 𝐴2 0 ]
𝐴 3 𝐷3∗ 𝐶3 𝐷1 [ 𝐷3 [ ∗ ∗ 0∗ 𝐵1 0 𝐸2 = 𝑟 [ 3 𝐵1 [𝐷1 𝐶 [ 𝐵3∗ 0 0 𝐶 0 0 [ 2 𝐴 3 𝐵3 [ 𝐵∗ [ ∗3 ∗ 0 𝐸3 = [ [𝐷1 𝐶3 0 [ 𝐶1 𝐷3 0 [ 𝐶2 𝐴 2
𝐶3∗
0 0 0 𝐵1∗ 𝐴1 0 0 0
𝐵3 0 0 0 0 𝐴2
𝐶3 0 0 0 0 0 𝐵1∗ 𝐴1 0 0 0
𝐶2∗ 𝐴∗2
𝐴 3 𝐵3 [ 𝐵∗ 0 [ 3 [ 𝐷3 0 𝐸5 = [ [𝐷1∗ 𝐶3∗ 0 [ [ 0 0 𝐴 𝐶 2 2 [
𝐷3∗
𝐶2∗ 𝐴∗2
𝐴3 [ 𝐵∗ [ 3∗ [ 𝐶 3 𝐸4 = [ [ 0 [ [𝐶1 𝐷3 [ 𝐶2
𝐷3∗
𝐶3 0 0 0 0 0 𝐴∗1 𝐴1 0 0 0
min 𝑖± [𝑓 (𝑋, 𝑌)] = 𝑟 (𝐸3 ) − 𝑟 [
0 𝐴∗1 0 0 0
(34)
𝑖± (𝐸2 ) − 𝑟 (𝐸5 )} .
(30)
Proof. By Lemma 1, the general solutions to (13) and (11) can be written as 𝑋 = 𝐴†2 𝐶2 + 𝐿 𝐴 2 𝑊,
] ] ] ], ] ] ]
𝑌 = 𝐴†1 𝐶1 + 𝐿 𝐴 1 𝐷1 𝐵1† + 𝐿 𝐴 1 𝑍𝑅𝐵1 ,
]
𝑄 = 𝐵3 𝐿 𝐴 2 ,
𝐶3 𝐷1 0 ] ] 𝐵1 ] ]. 0 ] ] 0 ] 0 ]
𝑇 = 𝐶3 𝐿 𝐴 1 ,
𝐽 = 𝑅𝐵1 𝐷3 ,
𝑃 = 𝐴 3 − 𝐵3 𝐴†2 𝐶2 − (𝐵3 𝐴†2 𝐶2 )
∗
(36)
− 𝐶3 (𝐴†1 𝐶1 + 𝐿 𝐴 1 𝐷1 𝐵1† ) 𝐷3 ∗
− (𝐶3 (𝐴†1 𝐶1 + 𝐿 𝐴 1 𝐷1 𝐵1† ) 𝐷3 ) .
(a) the maximal rank of (10) subject to (13) and (11) is max 𝑟 [𝑓 (𝑋, 𝑌)]
Substituting (36) into (10) yields 𝑓 (𝑋, 𝑌) = 𝑃 − 𝑄𝑊 − (𝑄𝑊)∗ − 𝑇𝑍𝐽 − (𝑇𝑍𝐽)∗ .
𝑋∈𝐺,𝑌∈𝑆
= min {𝑝, 𝑟 (𝐸1 ) − 2𝑟 (𝐴 1 ) − 2𝑟 (𝐴 2 ) ,
(35)
where 𝑊 and 𝑍 are arbitrary matrices with appropriate sizes. Put
Then one has the following:
(31)
𝑟 (𝐸2 ) − 2𝑟 (𝐵1 ) − 2𝑟 (𝐴 2 ) ,
(37)
Clearly 𝑃 is Hermitian. It follows from Lemma 6 that max 𝑟 [𝑓 (𝑋, 𝑌)]
𝑟 (𝐸3 ) − 2𝑟 (𝐴 2 ) − 𝑟 (𝐴 1 ) − 𝑟 (𝐵1 )} ,
𝑋∈𝐺,𝑌∈𝑆
= max 𝑟 (𝑃 − 𝑄𝑊 − (𝑄𝑊)∗ − 𝑇𝑍𝐽 − (𝑇𝑍𝐽)∗ )
(b) the minimal rank of (10) subject to (13) and (11) is
𝑊,𝑍
min 𝑟 [𝑓 (𝑋, 𝑌)]
𝑋∈𝐺,𝑌∈𝑆
= 2𝑟 (𝐸3 ) − 2𝑟 [
𝐵3 ] 𝐴2
+ max {𝑖± (𝐸1 ) − 𝑟 (𝐸4 ) ,
]
𝐷3∗ 𝐶1∗
0 0 0 0
0 0 0 0
], ] ]
𝑖± (𝐸2 ) − 𝑟 (𝐵1 ) − 𝑟 (𝐴 2 )} , (33)
𝑋∈𝐺,𝑌∈𝑆
𝐶2∗ 𝐴∗2 ] ] 0 0 0
max 𝑖± [𝑓 (𝑋, 𝑌)] = min {𝑖± (𝐸1 ) − 𝑟 (𝐴 1 ) − 𝑟 (𝐴 2 ) ,
𝑋∈𝐺,𝑌∈𝑆𝑥
(d) the minimal inertia of (10) subject to (13) and (11) is
𝐵3 𝐶2∗ 0 0] ] 0 0] , ∗] 0 𝐴 2] 𝐴2 0 ]
𝐷3∗
(c) the maximal inertia of (10) subject to (13) and (11) is
(38)
= min {𝑚, 𝑟 (𝑁1 ) , 𝑟 (𝑁2 ) , 𝑟 (𝑁3 )} , 𝐵3 ] 𝐴2
+ max {𝑟 (𝐸1 ) − 2𝑟 (𝐸4 ) , 𝑟 (𝐸2 ) − 2𝑟 (𝐸5 ) , 𝑖+ (𝐸1 ) + 𝑖− (𝐸2 ) − 𝑟 (𝐸4 ) − 𝑟 (𝐸5 ) , 𝑖− (𝐸1 ) + 𝑖+ (𝐸2 ) − 𝑟 (𝐸4 ) − 𝑟 (𝐸5 )} ,
min 𝑟 [𝑓 (𝑋, 𝑌)]
𝑋∈𝐺,𝑌∈𝑆
(32)
= max 𝑟 (𝑃 − 𝑄𝑊 − (𝑄𝑊)∗ − 𝑇𝑍𝐽 − (𝑇𝑍𝐽)∗ ) 𝑊,𝑍
= 2𝑟 (𝑁3 ) − 2𝑟 (𝑄) + max {𝑠+ + 𝑠− , 𝑡+ + 𝑡− , 𝑠+ + 𝑡− , 𝑠− + 𝑡+ } ,
(39)
Journal of Applied Mathematics
5
max 𝑖± [𝑓 (𝑋, 𝑌)]
By 𝐶1 𝐵1 = 𝐴 1 𝐷1 , we obtain
𝑋∈𝐺,𝑌∈𝑆
= max 𝑟 (𝑃 − 𝑄𝑊 − (𝑄𝑊)∗ − 𝑇𝑍𝐽 − (𝑇𝑍𝐽)∗ ) 𝑊,𝑍
(40)
𝑃 𝑄 𝐽∗
] [ 𝑟 (𝑁2 ) = 𝑟 [𝑄∗ 0 0 ]
= min {𝑖± (𝑁1 ) , 𝑖± (𝑁2 )} ,
[𝐽
min 𝑖± [𝑓 (𝑋, 𝑌)]
0 0] 𝐷3∗ 𝐶3 𝐷1 𝐵3 𝐶2∗
𝐴3
𝑋∈𝐺,𝑌∈𝑆
= max 𝑟 (𝑃 − 𝑄𝑊 − (𝑄𝑊)∗ − 𝑇𝑍𝐽 − (𝑇𝑍𝐽)∗ ) 𝑊,𝑍
(41)
= 𝑟 (𝑁3 ) − 𝑟 (𝑄) + max {𝑠± , 𝑡± } ,
[ [ 𝐷3 0 [ [𝐷∗ 𝐶∗ 𝐵∗ = 𝑟[ 1 3 1 [ [ 𝐵∗ 0 [ 3 0 [ 𝐶2
where
𝐵1 0 0 0
] 0] ] 0 0] ] ] ∗] 0 𝐴 2] 𝐴2 0 ] 0
− 2𝑟 (𝐵1 ) − 2𝑟 (𝐴 2 ) , 𝑃 𝑄 𝑇 𝑁1 = [𝑄∗ 0 0 ] , ∗ [𝑇 0 0 ]
𝑟 (𝑁3 ) = 𝑟 [
𝑃 𝑄 𝐽 𝑁2 = [𝑄∗ 0 0 ] , [ 𝐽 0 0] 𝑃 𝑄 𝑇 𝐽∗ 𝑁3 = [ ∗ ], 𝑄 0 0 0
(42)
∗
[𝑇
Now, we simplify the ranks and inertias of block matrices in (38)–(41). By Lemma 4, block Gaussian elimination, and noting that ∗
∗ †
= (𝐼 − 𝑆 𝑆) = 𝐼 − 𝑆 (𝑆 ) = 𝑅𝑆∗ ,
(43)
𝑃 𝑄 𝑇 𝑟 (𝑁1 ) = 𝑟 [𝑄∗ 0 0 ] ∗ [𝑇 0 0 ] 𝐶3 𝐷3∗ 𝐶1∗ 𝐵3 𝐶2∗ 0 𝐴∗1 0 0] ] 𝐴1 0 0 0] ] 0 0 0 𝐴∗2 ] 0 0 𝐴2 0 ]
− 2𝑟 (𝐴 1 ) − 2𝑟 (𝐴 2 ) .
0
0
0
0 𝐴1 𝐴2 0
] 0 𝐴∗2 ] ] 𝐵1∗ 0 ] ] ] 0 0] ] 0 0]
0 0 0]
𝐴3 [ 𝐵∗ [ 3∗ [ 𝐶 3 = 𝑟[ [ 0 [ [𝐶1 𝐷3 [ 𝐶2
𝐵3 0 0 0 0 𝐴2
𝐶3 𝐷3∗ 𝐶2∗ 𝐷3∗ 𝐶1∗ 0 0 𝐴∗2 0 ] ] 0 0 0 𝐴∗1 ] ] ∗ 0 𝐵1 0 0 ] ] 𝐴1 0 0 0 ] 0 0 0 0 ]
− 𝑟 (𝐵1 ) − 2𝑟 (𝐴 2 ) − 2𝑟 (𝐴 1 ) ,
𝑃 𝑄 𝑇 𝐽∗ 𝑟 (𝑁5 ) = 𝑟 [𝑄∗ 0 0 0 ] ∗ [𝑇 0 0 0 ] 𝐴3 [ 𝐵∗ [ 3 [ 𝐷3 = 𝑟[ [𝐷1∗ 𝐶3∗ [ [ 0 [ 𝐶2
we have the following:
𝐴3 [ 𝐶∗ [ 3 = 𝑟[ 3 [𝐶1 𝐷 [ 𝐵3∗ [ 𝐶2
0
[ ] 𝑟 (𝑁4 ) = 𝑟 [𝑄∗ 0 0 0 ]
𝑡± = 𝑖± (𝑁2 ) − 𝑟 (𝑁5 ) .
∗
𝐵3 𝐶3∗ 𝐷3∗ 𝐶2∗
𝑃 𝑄 𝑇 𝐽∗
𝑃 𝑄 𝑇 𝐽∗ [ 𝑁5 = 𝑄∗ 0 0 0 ] , [ 𝐽 0 0 0]
†
[ ∗ [ 𝐵3 [ [ = 𝑟 [𝐷1∗ 𝐶3∗ [ [𝐶 𝐷 [ 1 3 [ 𝐶2
]
− 𝑟 (𝐵1 ) − 2𝑟 (𝐴 2 ) − 𝑟 (𝐴 1 ) ,
𝑃 𝑄 𝑇 𝐽∗ 𝑁4 = [𝑄∗ 0 0 0 ] , ∗ [𝑇 0 0 0 ]
𝐿∗𝑆
𝑄∗ 0 0 0 𝐴3
∗
𝑠± = 𝑖± (𝑁1 ) − 𝑟 (𝑁4 ) ,
𝑃 𝑄 𝑇 𝐽∗
𝐵3 0 0 0 0 𝐴2
𝐶3 𝐷3∗ 0 0 0 0 0 𝐴∗1 𝐴1 0 0 0
𝐶2∗ 𝐶3 𝐷1 𝐴∗2 0 ] ] 0 𝐵1 ] ] 0 0 ] ] 0 0 ] 0 0 ]
− 2𝑟 (𝐵1 ) − 2𝑟 (𝐴 2 ) − 𝑟 (𝐴 1 ) . (44)
(45) By Lemma 2, we can get the following: 𝑃 𝑄 𝑇 𝑖± (𝑁1 ) = 𝑖± [𝑄∗ 0 0 ] ∗ [𝑇 0 0 ]
6
Journal of Applied Mathematics 𝐴3 [ 𝐶∗ [ 3 = 𝑖± [ 3 [𝐶1 𝐷 [ 𝐵3∗ [ 𝐶2
𝐶3 𝐷3∗ 𝐶1∗ 0 𝐴∗1 𝐴1 0 0 0 0 0
𝐵3 𝐶2∗ 0 0] ] 0 0] ] 0 𝐴∗2 ] 𝐴2 0 ]
(e) 𝐴 3 − 𝐵3 𝑋 − (𝐵3 𝑋)∗ − 𝐶3 𝑌𝐷3 − (𝐶3 𝑌𝐷3 )∗ > 0 for all 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 if and only if 𝑟 (𝐸3 ) − 𝑟 [
− 𝑟 (𝐴 1 ) − 𝑟 (𝐴 2 ) , (46) 𝑃 𝑄 𝐽∗ 𝑖± (𝑁2 ) = 𝑖± [𝑄∗ 0 0 ] [ 𝐽 0 0] 𝐴 3 𝐷3∗ 𝐶3 𝐷1 [ 𝐷3 [ ∗ ∗ 0∗ 𝐵1 0 = 𝑖± [ 3 𝐵1 [𝐷1 𝐶 [ 𝐵3∗ 0 0 𝐶 0 0 2 [
𝐵3 𝐶2∗ 0 0] ] 0 0] ] 0 𝐴∗2 ] 𝐴2 0 ]
(47)
Corollary 8. Let 𝐴 1 , 𝐶1 , 𝐵1 , 𝐷1 , 𝐴 2 , 𝐶2 , 𝐴 3 , 𝐵3 , 𝐶3 , 𝐷3 , and 𝐸𝑖 , (𝑖 = 1, 2, . . . , 5) be as in Theorem 7, and suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by 𝑆 and (11) by 𝐺. Then, one has the following: (a) there exist 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 such that 𝐴 3 − 𝐵3 𝑋 − (𝐵3 𝑋)∗ − 𝐶3 𝑌𝐷3 − (𝐶3 𝑌𝐷3 )∗ > 0 if and only if (48)
(b) there exist 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 such that 𝐴 3 − 𝐵3 𝑋 − (𝐵3 𝑋)∗ − 𝐶3 𝑌𝐷3 − (𝐶3 𝑌𝐷3 )∗ < 0 if and only if 𝑖− (𝐸1 ) − 𝑟 (𝐴 1 ) − 𝑟 (𝐴 2 ) ≥ 𝑝, 𝑖− (𝐸2 ) − 𝑟 (𝐵1 ) − 𝑟 (𝐴 2 ) ≥ 𝑝,
(49)
(c) there exist 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 such that 𝐴 3 − 𝐵3 𝑋 − (𝐵3 𝑋)∗ − 𝐶3 𝑌𝐷3 − (𝐶3 𝑌𝐷3 )∗ ≥ 0 if and only if 𝑟 (𝐸3 ) − 𝑟 [ 𝑟 (𝐸3 ) − 𝑟 [
𝐵3 ] + 𝑖− (𝐸1 ) − 𝑟 (𝐸4 ) ≤ 0, 𝐴2 𝐵3 ] + 𝑖− (𝐸2 ) − 𝑟 (𝐸5 ) ≤ 0. 𝐴2
𝐵3 ] + 𝑖+ (𝐸1 ) − 𝑟 (𝐸4 ) ≤ 0, 𝐴2
𝐵 𝑟 (𝐸3 ) − 𝑟 [ 3 ] + 𝑖+ (𝐸2 ) − 𝑟 (𝐸5 ) ≤ 0, 𝐴2
𝐵3 ] + 𝑖− (𝐸1 ) − 𝑟 (𝐸4 ) = 𝑝 𝐴2
(52)
(50)
(51)
(53)
(g) 𝐴 3 − 𝐵3 𝑋 − (𝐵3 𝑋)∗ − 𝐶3 𝑌𝐷3 − (𝐶3 𝑌𝐷3 )∗ ≥ 0 for all 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 if and only if 𝑖− (𝐸1 ) − 𝑟 (𝐴 1 ) − 𝑟 (𝐴 2 ) ≤ 0 𝑜𝑟 𝑖− (𝐸2 ) − 𝑟 (𝐵1 ) − 𝑟 (𝐴 2 ) ≤ 0,
(54)
(h) 𝐴 3 − 𝐵3 𝑋 − (𝐵3 𝑋)∗ − 𝐶3 𝑌𝐷3 − (𝐶3 𝑌𝐷3 )∗ ≤ 0 for all 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 if and only if 𝑖+ (𝐸1 ) − 𝑟 (𝐴 1 ) − 𝑟 (𝐴 2 ) ≤ 0 𝑜𝑟 𝑖+ (𝐸2 ) − 𝑟 (𝐵1 ) − 𝑟 (𝐴 2 ) ≤ 0,
(55)
(i) there exist 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 such that 𝐴 3 − 𝐵3 𝑋 − (𝐵3 𝑋)∗ −𝐶3 𝑌𝐷3 −(𝐶3 𝑌𝐷3 )∗ is nonsingular if and only if 𝑟 (𝐸1 ) − 2𝑟 (𝐴 1 ) − 2𝑟 (𝐴 2 ) ≥ 𝑝, 𝑟 (𝐸2 ) − 2𝑟 (𝐵1 ) − 2𝑟 (𝐴 2 ) ≥ 𝑝,
(d) there exist 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 such that 𝐴 3 − 𝐵3 𝑋 − (𝐵3 𝑋)∗ − 𝐶3 𝑌𝐷3 − (𝐶3 𝑌𝐷3 )∗ ≤ 0 if and only if 𝑟 (𝐸3 ) − 𝑟 [
𝑟 (𝐸3 ) − 𝑟 [
𝐵 𝑜𝑟 𝑟 (𝐸3 ) − 𝑟 [ 3 ] + 𝑖− (𝐸2 ) − 𝑟 (𝐸5 ) = 𝑝, 𝐴2
Substituting (44)-(47) into (38) and (41) yields (31)–(34), respectively.
𝑖+ (𝐸2 ) − 𝑟 (𝐵1 ) − 𝑟 (𝐴 2 ) ≥ 𝑝.
𝐵 𝑜𝑟 𝑟 (𝐸3 ) − 𝑟 [ 3 ] + 𝑖+ (𝐸2 ) − 𝑟 (𝐸5 ) = 𝑝, 𝐴2
(f) 𝐴 3 − 𝐵3 𝑋 − (𝐵3 𝑋)∗ − 𝐶3 𝑌𝐷3 − (𝐶3 𝑌𝐷3 )∗ < 0 for all 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 if and only if
− 𝑟 (𝐵1 ) − 𝑟 (𝐴 2 ) .
𝑖+ (𝐸1 ) − 𝑟 (𝐴 1 ) − 𝑟 (𝐴 2 ) ≥ 𝑝,
𝐵3 ] + 𝑖+ (𝐸1 ) − 𝑟 (𝐸4 ) = 𝑝 𝐴2
(56)
𝑟 (𝐸3 ) − 2𝑟 (𝐴 2 ) − 𝑟 (𝐴 1 ) − 𝑟 (𝐵1 ) ≥ 𝑝.
3. Relations between the Hermitian Part of the Solutions to (13) and (11) Now we consider the extremal ranks and inertias of the difference between the Hermitian part of the solutions to (13) and (11). Theorem 9. Let 𝐴 1 ∈ C𝑚×𝑝 , 𝐶1 ∈ C𝑚×𝑝 , 𝐵1 ∈ C𝑝×𝑙 , 𝐷1 ∈ C𝑝×𝑙 , 𝐴 2 ∈ C𝑡×𝑝 , and 𝐶2 ∈ C𝑡×𝑝 , be given. Suppose that the system of matrix equations (13) and (11) is consistent,
Journal of Applied Mathematics
7
respectively. Denote the set of all solutions to (13) by 𝑆 and (11) by 𝐺. Put 0 [𝐼 [ 𝐻1 = [ [𝐶1 [−𝐼 [𝐶2
𝐼 0 𝐴1 0 0
𝐶1∗ 𝐴∗1
= 𝑟 (𝐸3 ) − 𝑝 + max {𝑖± (𝐻1 ) − 𝑟 (𝐻4 ) , 𝑖± (𝐻2 ) − 𝑟 (𝐻5 )} . (58)
𝐶2∗
−𝐼 0 0] ] 0 0 0] ], 0 0 𝐴∗2 ] 0 𝐴2 0 ]
Proof. By letting 𝐴 3 = 0, 𝐵3 = −𝐼, 𝐶3 = 𝐼, and 𝐷3 = 𝐼 in Theorem 7, we can get the results. Corollary 10. Let 𝐴 1 , 𝐶1 , 𝐵1 , 𝐷1 , 𝐴 2 , 𝐶2 , and 𝐻𝑖 , (𝑖 = 1, 2, . . . , 5) be as in Theorem 9, and suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by 𝑆 and (11) by 𝐺. Then, one has the following:
𝐶2∗
0 [ 𝐼 [ ∗ 𝐻2 = 𝑟 [ [𝐷1 [ −𝐼 [ 𝐶2
𝐼 0 𝐵1∗ 0 0
𝐷1 𝐵1 0 0 0
−𝐼 0 0] ] 0 0] ], 0 𝐴∗2 ] 𝐴2 0 ]
0 [ −𝐼 [ ∗ 𝐻3 = [ [𝐷1 [ 𝐶1 [ 𝐶2
−𝐼 0 0 0 𝐴2
𝐼 0 0 𝐴1 0
𝐼 𝐶2∗ 0 𝐴∗2 ] ] 𝐵1∗ 0 ] ], 0 0] 0 0]
𝐼 0 0 𝐵1∗ 0 0
𝐶2∗ 𝐴∗2
0 [−𝐼 [ [𝐼 𝐻4 = [ [0 [ [𝐶1 [𝐶2
−𝐼 0 0 0 0 𝐴2
𝐼 0 0 0 𝐴1 0
0 [ −𝐼 [ [ 𝐼 𝐻5 = [ [𝐷1∗ [ [0 [ 𝐶2
−𝐼 0 0 0 0 𝐴2
𝐼 𝐼 𝐶2∗ 0 0 𝐴∗2 0 0 0 0 𝐴∗1 0 𝐴1 0 0 0 0 0
(a) there exist 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 such that (𝑋 + 𝑋∗ ) > (𝑌 + 𝑌∗ ) if and only if 𝑖+ (𝐻1 ) − 𝑟 (𝐴 1 ) − 𝑟 (𝐴 2 ) ≥ 𝑝, (57)
𝐶1∗
0] ] 0 𝐴∗1 ] ], 0 0] ] 0 0] 0 0] 𝐷1 0] ] 𝐵1 ] ]. 0] ] 0] 0]
Then one has the following: max 𝑟 [(𝑋 + 𝑋∗ ) − (𝑌 + 𝑌∗ )]
𝑋∈𝐺,𝑌∈𝑆
= min {𝑝, 𝑟 (𝐻1 ) − 2𝑟 (𝐴 1 ) − 2𝑟 (𝐴 2 ) , 𝑟 (𝐻2 ) − 2𝑟 (𝐵1 ) −2𝑟 (𝐴 2 ) , 𝑟 (𝐻3 ) − 2𝑟 (𝐴 2 ) − 𝑟 (𝐴 1 ) − 𝑟 (𝐵1 )} , min 𝑟 [(𝑋 + 𝑋∗ ) − (𝑌 + 𝑌∗ )]
𝑋∈𝐺,𝑌∈𝑆
= 2𝑟 (𝐻3 ) − 2𝑝 + max {𝑟 (𝐻1 ) − 2𝑟 (𝐻4 ) , 𝑟 (𝐻2 ) − 2𝑟 (𝐻5 ) , 𝑖+ (𝐻1 ) + 𝑖− (𝐻2 ) − 𝑟 (𝐻4 ) − 𝑟 (𝐻5 ) , 𝑖− (𝐻1 ) + 𝑖+ (𝐻2 ) − 𝑟 (𝐻4 ) − 𝑟 (𝐻5 )} , max 𝑖± [(𝑋 + 𝑋∗ ) − (𝑌 + 𝑌∗ )]
𝑋∈𝐺,𝑌∈𝑆
= min {𝑖± (𝐻1 ) − 𝑟 (𝐴 1 ) − 𝑟 (𝐴 2 ) , 𝑖± (𝐻2 ) − 𝑟 (𝐵1 ) − 𝑟 (𝐴 2 )} ,
min 𝑖± [(𝑋 + 𝑋∗ ) − (𝑌 + 𝑌∗ )]
𝑋∈𝐺,𝑌∈𝑆
𝑖+ (𝐻2 ) − 𝑟 (𝐵1 ) − 𝑟 (𝐴 2 ) ≥ 𝑝.
(59)
(b) there exist 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 such that (𝑋 + 𝑋∗ ) < (𝑌 + 𝑌∗ ) if and only if 𝑖− (𝐻1 ) − 𝑟 (𝐴 1 ) − 𝑟 (𝐴 2 ) ≥ 𝑝, 𝑖− (𝐻2 ) − 𝑟 (𝐵1 ) − 𝑟 (𝐴 2 ) ≥ 𝑝,
(60)
(c) there exist 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 such that (𝑋 + 𝑋∗ ) ≥ (𝑌 + 𝑌∗ ) if and only if 𝑟 (𝐻3 ) − 𝑝 + 𝑖− (𝐻1 ) − 𝑟 (𝐻4 ) ≤ 0, 𝑟 (𝐻3 ) − 𝑝 + 𝑖− (𝐻1 ) − 𝑟 (𝐻4 ) ≤ 0,
(61)
(d) there exist 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 such that (𝑋 + 𝑋∗ ) ≤ (𝑌 + 𝑌∗ ) if and only if 𝑟 (𝐻3 ) − 𝑝 + 𝑖+ (𝐻1 ) − 𝑟 (𝐻4 ) ≤ 0, 𝑟 (𝐻3 ) − 𝑝 + 𝑖+ (𝐻2 ) − 𝑟 (𝐻5 ) ≤ 0,
(62)
(e) (𝑋 + 𝑋∗ ) > (𝑌 + 𝑌∗ ) for all 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 if and only if 𝑟 (𝐻3 ) − 𝑝 + 𝑖+ (𝐻1 ) − 𝑟 (𝐻4 ) = 𝑝 𝑜𝑟 𝑟 (𝐻3 ) − 𝑝 + 𝑖+ (𝐻2 ) − 𝑟 (𝐻5 ) = 𝑝,
(63)
(f) (𝑋 + 𝑋∗ ) < (𝑌 + 𝑌∗ ) for all 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 if and only if 𝑟 (𝐻3 ) − 𝑝 + 𝑖− (𝐻1 ) − 𝑟 (𝐻4 ) = 𝑝 𝑜𝑟 𝑟 (𝐻3 ) − 𝑝 + 𝑖− (𝐻2 ) − 𝑟 (𝐻5 ) = 𝑝,
(64)
(g) (𝑋 + 𝑋∗ ) ≥ (𝑌 + 𝑌∗ ) for all 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 if and only if 𝑖− (𝐻1 ) − 𝑟 (𝐴 1 ) − 𝑟 (𝐴 2 ) ≤ 0 𝑜𝑟 𝑖− (𝐻2 ) − 𝑟 (𝐵1 ) − 𝑟 (𝐴 2 ) ≤ 0,
(65)
8
Journal of Applied Mathematics (h) (𝑋 + 𝑋∗ ) ≤ (𝑌 + 𝑌∗ ) for all 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 if and only if 𝑖+ (𝐻1 ) − 𝑟 (𝐴 1 ) − 𝑟 (𝐴 2 ) ≤ 0
(66)
𝑜𝑟 𝑖+ (𝐻2 ) − 𝑟 (𝐵1 ) − 𝑟 (𝐴 2 ) ≤ 0, ∗
(i) there exist 𝑋 ∈ 𝐺 and 𝑌 ∈ 𝑆 such that (𝑋 + 𝑋 ) − (𝑌 + 𝑌∗ ) is nonsingular if and only if
where 𝑈1 , 𝑈2 , 𝑉1 , 𝑉2 , 𝑊1 , and 𝑊2 are arbitrary matrices over C with appropriate sizes. Now we give the main theorem of this section. Theorem 12. Let 𝐴 𝑖 , 𝐶𝑖 , (𝑖 = 1, 2, 3), 𝐵𝑗 , and 𝐷𝑗 , (𝑗 = 1, 3) be given. Set 𝐴 = 𝐵3 𝐿 𝐴 2 ,
𝑟 (𝐻1 ) − 2𝑟 (𝐴 1 ) − 2𝑟 (𝐴 2 ) ≥ 𝑝, 𝑟 (𝐻2 ) − 2𝑟 (𝐵1 ) − 2𝑟 (𝐴 2 ) ≥ 𝑝,
𝐶 = 𝑅𝐵1 𝐷3 ,
(67)
𝐺 = 𝐶𝑅𝐴 ,
𝑟 (𝐻3 ) − 2𝑟 (𝐴 2 ) − 𝑟 (𝐴 1 ) − 𝑟 (𝐵1 ) ≥ 𝑝.
𝑁 = 𝐹∗ 𝐿 𝐺,
4. The Solvability Conditions and the General Solution to System (6)
(1) equation (5) is consistent, (2) 𝑅𝑀𝑅𝐴𝐸 = 0,
𝐿 𝐵 𝐸𝐿 𝐵 = 0,
− 𝐶3 (𝐴†1 𝐶1 + 𝐿 𝐴 1 𝐷1 𝐵1† ) 𝐷3
(69)
𝐸1 𝐶1∗ 𝐴 1 𝑟 [𝐴∗1 0 0 ] = 2𝑟 [𝐶1∗ 𝐴 1 ] . [ 𝐶1 0 0 ] In this case, the general solution of (5) can be expressed as ∗ 1 𝑌 = [𝐴† 𝐸𝐵† − 𝐴† 𝐵∗ 𝑀† 𝐸𝐵† − 𝐴† 𝑆(𝐵† ) 𝐸𝑁† 𝐴∗ 𝐵† 2 ∗
(72)
∗
− 𝐷3∗ (𝐴†1 𝐶1 + 𝐿 𝐴 1 𝐷1 𝐵1† ) 𝐶3∗ , 𝐸 = 𝑅𝐴 𝐷𝑅𝐴 .
(73)
Then the following statements are equivalent: (1) system (6) is consistent, (2) the equalities in (14) and (17) hold, and 𝑅𝐹 𝐸𝑅𝐹 = 0,
𝐿 𝐺𝐸𝐿 𝐺 = 0,
(74)
(3) the equalities in (15) and (18) hold, and
𝐸1 𝐵1 𝐶1∗ 𝐴 1 ] = 𝑟 [𝐵1 𝐶1∗ 𝐴 1 ] + 𝑟 (𝐴 1 ) , 𝐴∗1 0 0 0 𝐸1 𝐵1 𝐴 1 𝑟 [𝐴∗1 0 0 ] = 2𝑟 [𝐵1 𝐴 1 ] , ∗ [ 𝐵1 0 0 ]
𝑆 = 𝐺∗ 𝐿 𝑀,
(68)
(3) 𝑟[
(71)
𝑀 = 𝑅𝐹 𝐺∗ , ∗
𝑅𝑀𝑅𝐹 𝐸 = 0,
𝑅𝐴 𝐸𝑅𝐴 = 0,
𝐹 = 𝑅𝐴 𝐵,
𝐷 = 𝐴 3 − 𝐵3 𝐴†2 𝐶2 − (𝐵3 𝐴†2 𝐶2 )
We now turn our attention to (6). We in this section use Theorem 9 to give some necessary and sufficient conditions for the existence of a solution to (6) and present an expression of the general solution to (6). We begin with a lemma which is used in the latter part of this section. Lemma 11 (see [14]). Let 𝐴 1 ∈ C𝑚×𝑛1 , 𝐵1 ∈ C𝑚×𝑛2 , 𝐶1 ∈ be given. Let 𝐴 = 𝑅𝐴 1 𝐵1 , 𝐵 = 𝐶1 𝑅𝐴 1 , C𝑞×𝑚 , and 𝐸1 ∈ C𝑚×𝑚 ℎ 𝐸 = 𝑅𝐴 1 𝐸1 𝑅𝐴 1 , 𝑀 = 𝑅𝐴 𝐵∗ , 𝑁 = 𝐴∗ 𝐿 𝐵 , and 𝑆 = 𝐵∗ 𝐿 𝑀. Then the following statements are equivalent:
𝐵 = 𝐶3 𝐿 𝐴 1 ,
∗
+𝐴† 𝐸(𝑀† ) + (𝑁† ) 𝐸𝐵† 𝑆† 𝑆] + 𝐿 𝐴𝑉1 + 𝑉2 𝑅𝐵 + 𝑈1 𝐿 𝑆 𝐿 𝑀 + 𝑅𝑁𝑈2∗ 𝐿 𝑀 − 𝐴† 𝑆𝑈2 𝑅𝑁𝐴∗ 𝐵† ,
𝐴3 [ 𝐵∗ [ 3 𝑟[ [ 𝐶1∗𝐷3∗ [𝐷1 𝐶3 [ 𝐶2
𝐶3 𝐷3∗ 0 0 𝐴1 0 0 𝐵1∗ 0 0
𝐵3 𝐶2∗ 𝐶3 𝐷3∗ 0 𝐴∗2 ] ] [𝐴 1 0 [ 0 0] ] = 𝑟 [ 0 𝐵1∗ 0 0] [0 0 𝐴2 0 ]
𝐵3 0] ] + 𝑟 [𝐴 2 ] , 0] 𝐵3 𝐴 2]
𝐶3 0 0 𝐴1 0
𝐵3 𝐷3∗ 𝐶1∗ 𝐶2∗ 0 𝐴∗1 0] 𝐶3 𝐵3 ] [𝐴 1 0 ] , 0 0 𝐴∗2 ] = 2𝑟 ] 0 0 0] [ 0 𝐴 2] 𝐴2 0 0]
𝐴 3 𝐷3∗ [ 𝐷3 0 [ ∗ 𝑟[ [ 𝐵∗ 3 ∗ 0∗ [𝐷1 𝐶3 𝐵1 0 [ 𝐶2
𝐵3 𝐶3 𝐷1 𝐶2∗ 0 𝐵1 0] 𝐷3∗ 𝐵3 ] ∗ ] [ 0 0 𝐴∗2 ] ] = 2𝑟 𝐵1 0 . 0 0 0] 0 𝐴 2] [ 𝐴2 0 0]
𝐴3 [ 𝐶∗ [ ∗3 𝑟[ [ 𝐵3 [𝐶1 𝐷3 [ 𝐶2
(75)
∗
𝑋 = 𝐴†1 [𝐸1 − 𝐵1 𝑌𝐶1 − (𝐵1 𝑌𝐶1 ) ]
In this case, the general solution of system (6) can be expressed as
1 ∗ − 𝐴†1 [𝐸1 − 𝐵1 𝑌𝐶1 − (𝐵1 𝑌𝐶1 ) ] 𝐴 1 𝐴†1 2 −
𝐴†1 𝑊1 𝐴∗1
+
𝑊1∗ 𝐴 1 𝐴†1
+ 𝐿 𝐴 1 𝑊2 ,
𝑋 = 𝐴†2 𝐶2 + 𝐿 𝐴 2 𝑈, (70)
𝑌 = 𝐴†1 𝐶1 + 𝐿 𝐴 1 𝐷1 𝐵1† + 𝐿 𝐴 1 𝑉𝑅𝐵1 ,
(76)
Journal of Applied Mathematics
9 𝐷3∗ 𝐵3 𝐶3 𝐷1 𝐶2∗
𝐴3
where ∗ 1 𝑉 = [𝐹† 𝐸𝐺† − 𝐹† 𝐺∗ 𝑀† 𝐸𝐺† − 𝐹† 𝑆(𝐺† ) 𝐸𝑁† 𝐹∗ 𝐺† 2 ∗
∗
+𝐹† 𝐸(𝑀† ) + (𝑁† ) 𝐸𝐺† 𝑆† 𝑆] + 𝐿 𝐹 𝑉1 + 𝑉2 𝑅𝐺 + 𝑈1 𝐿 𝑆 𝐿 𝑀 + 𝑅𝑁𝑈2∗ 𝐿 𝑀 − 𝐹† 𝑆𝑈2 𝑅𝑁𝐹∗ 𝐺† , †
∗
[ [ 𝐷3 0 0 [ [ 𝐵∗ 0 0 𝐿 𝐺𝐸𝐿 𝐺 = 0 ⇐⇒ 𝑟 [ 3 [ ∗ ∗ [𝐷 𝐶 𝐵∗ 0 [ 1 3 1 0 𝐴2 [ 𝐶2
1 − 𝐴† [𝐷 − 𝐵𝑉𝐶 − (𝐵𝑉𝐶)∗ ] 𝐴𝐴† 2 †
∗
− 𝐴 𝑊1 𝐴 +
𝑊1∗ 𝐴𝐴†
+ 𝐿 𝐴𝑊2 , (77)
(1) ⇔ (2): We separate the four equations in system (6) into three groups: 𝐴 1 𝑌 = 𝐶1 ,
= 𝑟 [𝐵 𝐶∗ 𝐴] + 𝑟 (𝐴)
𝐶3 𝐷3∗ [𝐴 1 0 = 𝑟[ [ 0 𝐵1∗ [0 0
𝐴 2 𝑋 = 𝐶2 ,
𝐶3 𝐷3∗ 0 0 𝐴1 0 0 𝐵1∗ 0 0
𝐶3 𝐵3 = 2𝑟 [𝐴 1 0 ] , [ 0 𝐴 2]
∗
𝑅𝑀𝑅𝐹 𝐸 = 0, (78)
𝐵3 𝐶2∗ 0 𝐴∗2 ] ] 0 0] ] 0 0] 𝐴2 0 ]
∗
𝐵3 0 0 0 𝐴2
(82)
(83)
𝑅𝐹 𝐸𝑅𝐹 = 0,
𝐿 𝐺𝐸𝐿 𝐺 = 0.
(84)
We know by Lemma 11 that the general solution of (83) can be expressed as (77). In Theorem 12, let 𝐴 1 and 𝐷1 vanish. Then we can obtain the general solution to the following system: 𝐴 2 𝑋 = 𝐶2 ,
𝑌𝐵1 = 𝐷1 ,
∗
∗
𝐵3 𝑋 + (𝐵3 𝑋) + 𝐶3 𝑌𝐷3 + (𝐶3 𝑌𝐷3 ) = 𝐴 3 .
(85)
Corollary 13. Let 𝐴 2 , 𝐶2 , 𝐵1 , 𝐷1 , 𝐵3 , 𝐶3 , 𝐷3 , and 𝐴 3 = 𝐴∗3 be given. Set 𝐴 = 𝐵3 𝐿 𝐴 2 ,
𝐶 = 𝑅𝐵1 𝐷3 ,
𝐹 = 𝑅𝐴 𝐶3 ,
By a similar approach, we can obtain that 𝐶3 0 0 𝐴1 0
(81)
Hence, the system (5) is consistent if and only if (80), (81), and (83) are consistent, respectively. It follows from Lemma 11 that (83) is solvable if and only if
𝐵3 0] ] + 𝑟 [𝐴 2 ] . 0] 𝐵3 𝐴 2]
𝐴3 [ 𝐶∗ [ ∗3 𝑅𝐹 𝐸𝑅𝐹 = 0 ⇐⇒ 𝑟 [ [ 𝐵3 [𝐶1 𝐷3 [ 𝐶2
(80)
𝐴𝑈 + (𝐴𝑈)∗ + 𝐵𝑉𝐶 + (𝐵𝑉𝐶)∗ = 𝐷.
𝐵3 0 0 𝐴∗2 ] ] 0 0] ] 0 0] 𝐴2 0 ]
𝐵3 0] ] + 𝑟 [𝐴 2 ] 0] 𝐵3 𝐴 2]
𝐴3 [ 𝐵∗ [ 3 ⇐⇒ 𝑟 [ [ 𝐶1∗𝐷3∗ [𝐷1 𝐶3 [ 𝐶2
𝑌𝐵1 = 𝐷1 ,
By Lemma 1, we obtain that system (80) is solvable if and only if (14), (81) is consistent if and only if (17). The general solutions to system (80) and (81) can be expressed as (16) and (19), respectively. Substituting (16) and (19) into (82) yields
𝐷 𝐵 𝐶∗ 𝐴 ] 𝐴∗ 0 0 0
𝐶3 𝐷3∗ [𝐴 1 0 = 𝑟[ [ 0 𝐵1∗ [0 0
0] ].
𝐵3 𝑋 + (𝐵3 𝑋) + 𝐶3 𝑌𝐷3 + (𝐶3 𝑌𝐷3 ) = 𝐴 3 .
𝑅𝑀𝑅𝐹 𝐸 = 0 ⇐⇒ 𝑟 (𝑅𝑀𝑅𝐹 𝐸)
𝐶3 𝐷3∗ 0 0 𝐴1 0 0 𝐵1∗ 0 0
0
(79)
Proof. (2) ⇔ (3): Applying Lemma 3 and Lemma 11 gives
𝐷 [𝐵∗ [ 3 ⇐⇒ 𝑟 [ [0 [0 [0
0
[ 0 𝐴 2]
where 𝑈1 , 𝑈2 , 𝑉1 , 𝑉2 , 𝑊1 , and 𝑊2 are arbitrary matrices over C with appropriate sizes.
= 0 ⇐⇒ 𝑟 [
0
𝐷3∗ 𝐵3
[ = 2𝑟 [ 𝐵1∗
𝑈 = 𝐴 [𝐷 − 𝐵𝑉𝐶 − (𝐵𝑉𝐶) ]
] 0] ] 𝐴∗2 ] ] ] 0] ] 0]
𝐵1
𝐷3∗ 𝐶1∗ 𝐴∗1 0 0 0
𝐶2∗
0] ] 𝐴∗2 ] ] 0] 0]
𝐺 = 𝐶𝑅𝐴 ,
𝑀 = 𝑅𝐹 𝐺∗ ,
𝑁 = 𝐹∗ 𝐿 𝐺 ,
𝑆 = 𝐺∗ 𝐿 𝑀, 𝐷 = 𝐴3 −
𝐵3 𝐴†2 𝐶2
−
(86)
∗ (𝐵3 𝐴†2 𝐶2 ) ∗
− 𝐶3 𝐷1 𝐵1† 𝐷3 − (𝐶3 𝐷1 𝐵1† 𝐷3 ) , 𝐸 = 𝑅𝐴 𝐷𝑅𝐴 .
10
Journal of Applied Mathematics
Then the following statements are equivalent: (1) system (85) is consistent (2) 𝑅𝐴 2 𝐶2 = 0,
𝐷1 𝐿 𝐵1 = 0,
𝑅𝐹 𝐸𝑅𝐹 = 0,
𝑅𝑀𝑅𝐹 𝐸 = 0,
𝐿 𝐺𝐸𝐿 𝐺 = 0,
(87) (88)
(3) 𝑟 [𝐴 2 𝐶2 ] = 𝑟 (𝐴 2 ) , 𝐴3 [ 𝐵∗ [ 𝑟 [ ∗3 ∗ 𝐷1 𝐶3 [ 𝐶2
𝐶3 𝐷3∗ 0 0 0 𝐵1∗ 0 0 𝐴3 [𝐶∗ 3 𝑟[ [ 𝐵3∗ [ 𝐶2
[
𝐷1 ] = 𝑟 (𝐵1 ) , 𝐵1
𝐵3 𝐶2∗ 𝐶3 𝐷3∗ 𝐵3 0 𝐴∗2 ] ] = 𝑟 [ 0 𝐵1∗ 0 ] + 𝑟 [𝐴 2 ] , 0 0] 𝐵3 [ 0 0 𝐴 2] 𝐴2 0 ] 𝐶3 𝐵3 𝐶2∗ 0 0 0] ] = 2𝑟 [𝐶3 𝐵3 ] , 0 0 𝐴∗2 ] 0 𝐴2 0 𝐴2 0 ]
𝐴 3 𝐷3∗ [ 𝐷3 0 [ ∗ 𝐵 𝑟[ 3 [ ∗ ∗ 0∗ [𝐷1 𝐶3 𝐵1 0 [ 𝐶2
𝐵3 𝐶3 𝐷1 𝐶2∗ 0 𝐵1 0] 𝐷3∗ 𝐵3 ∗] ] [ 0 0 𝐴 2 ] = 2𝑟 𝐵1∗ 0 ] . 0 0 0] [ 0 𝐴 2] 𝐴2 0 0]
(89)
In this case, the general solution of system (6) can be expressed as 𝑋 = 𝐴†2 𝐶2 + 𝐿 𝐴 2 𝑈, 𝑌 = 𝐷1 𝐵1† + 𝑉𝑅𝐵1 ,
(90)
where 𝑉=
∗ 1 † † [𝐹 𝐸𝐺 − 𝐹† 𝐺∗ 𝑀† 𝐸𝐺† − 𝐹† 𝑆(𝐺† ) 𝐸𝑁† 𝐹∗ 𝐺† 2 ∗
∗
+𝐹† 𝐸(𝑀† ) + (𝑁† ) 𝐸𝐺† 𝑆† 𝑆] + 𝐿 𝐹 𝑉1 + 𝑉2 𝑅𝐺 + 𝑈1 𝐿 𝑆 𝐿 𝑀 + 𝑅𝑁𝑈2∗ 𝐿 𝑀 − 𝐹† 𝑆𝑈2 𝑅𝑁𝐹∗ 𝐺† , ∗
𝑈 = 𝐴† [𝐷 − 𝐶3 𝑉𝐶 − (𝐶3 𝑉𝐶) ] 1 ∗ − 𝐴† [𝐷 − 𝐶3 𝑉𝐶 − (𝐶3 𝑉𝐶) ] 𝐴𝐴† 2 − 𝐴† 𝑊1 𝐴∗ + 𝑊1∗ 𝐴𝐴† + 𝐿 𝐴𝑊2 ,
(91)
where 𝑈1 , 𝑈2 , 𝑉1 , 𝑉2 , 𝑊1 , and 𝑊2 are arbitrary matrices over C with appropriate sizes.
Acknowledgments The authors would like to thank Dr. Mohamed, Dr. Sivakumar, and a referee very much for their valuable suggestions
and comments, which resulted in a great improvement of the original paper. This research was supported by the Grants from the National Natural Science Foundation of China (NSFC (11161008)) and Doctoral Program Fund of Ministry of Education of P.R.China (20115201110002).
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