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; zxjnsc@163.com 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 π(π, π) = π΄ 3 − π΅3 π − (π΅3 π)∗ − πΆ3 ππ·3 − (πΆ3 ππ·3 )∗ , where π΄ 3 = π΄∗3 , π΅3 , πΆ3 , and π·3 are known matrices and π and π are the solutions to the matrix equations π΄ 1 π = πΆ1 , ππ΅1 = π·1 , and π΄ 2 π = πΆ2 , respectively. As applications, we present necessary and sufficient condition for the previous matrix function π(π, π) 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 π΄ 1 π = πΆ1 , ππ΅1 = π·1 , π΄ 2 π = πΆ2 , and π΅3 π + (π΅3 π)∗ + πΆ3 ππ·3 + (πΆ3 ππ·3 )∗ = π΄ 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 π × π matrices over C by Cπ×π , and the set of . The symbols π΄∗ and all π × π Hermitian matrices by Cπ×π β R(π΄) stand for the conjugate transpose, the column space of a complex matrix π΄ respectively. πΌπ denotes the π × π identity matrix. The Moore-Penrose inverse [1] 𴆠of π΄, is the unique solution π to the four matrix equations: (i) π΄ππ΄ = π΄, (ii) ππ΄π = π, (iii) (π΄π)∗ = π΄π, symbols π+ (π΄) and π− (π΄) are called the positive index and the negative index of inertia, respectively. For two Hermitian matrices π΄ and π΅ of the same sizes, we say π΄ ≥ π΅ (π΄ ≤ π΅) in the LoΜwner partial ordering if π΄ − π΅ is positive (negative) semidefinite. The Hermitian part of π is defined as π»(π) = π + π∗ . We will say that π is Re-nnd (Re-nonnegative semidefinite) if π»(π) ≥ 0, π is Re-pd (Re-positive definite) if π»(π) > 0, and π is Re-ns if π»(π) 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) π΅π + (π΅π)∗ = π΄. (3) ∗ (iv) (ππ΄) = ππ΄. Moreover, πΏ π΄ and π π΄ stand for the projectors πΏ π΄ = πΌ − 𴆠π΄, π π΄ = πΌ − π΄π΄† induced by π΄. It is well known that the eigenvalues of a Hermitian matrix π΄ ∈ Cπ×π are real, and the inertia of π΄ is defined to be the triplet Iπ (π΄) = {π+ (π΄) , π− (π΄) , π0 (π΄)} , (2) where π+ (π΄), π− (π΄), and π0 (π΄) stand for the numbers of positive, negative, and zero eigenvalues of π΄, respectively. The In 2007, DjordjevicΜ [11] considered the explicit solution to (3) for linear bounded operators on Hilbert spaces. Moreover, Cao [12] investigated the general explicit solution to π΅ππΆ + (π΅ππΆ)∗ = π΄. (4) Xu et al. [13] obtained the general expression of the solution of operator equation (4). In 2012, Wang and He [14] studied 2 Journal of Applied Mathematics some necessary and sufficient conditions for the consistence of the matrix equation ∗ ∗ π΄ 1 π + (π΄ 1 π) + π΅1 ππΆ1 + (π΅1 ππΆ1 ) = πΈ1 (5) and presented an expression of the general solution to (5). Note that (5) is a special case of the following system: π΄ 1 π = πΆ1 , ππ΅1 = π·1 , π΄ 2 π = πΆ2 , ∗ (6) ∗ π΅3 π + (π΅3 π) + πΆ3 ππ·3 + (πΆ3 ππ·3 ) = π΄ 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 π΄ 1 , πΆ1 , π΅1 , and π·1 be given. Then the following statements are equivalent: (1) system (13) is consistent, (2) π π΄ 1 πΆ1 = 0, π·1 πΏ π΅1 = 0, π΄ 1 π·1 = πΆ1 π΅1 . (3) π [π΄ 1 πΆ1 ] = π (π΄ 1 ) , [ π·1 ] = π (π΅1 ) , π΅1 (7) π (π) = π΄ − π΅ππΆ − (π΅ππΆ)∗ , (8) In this case, the general solution can be written as β (π, π) = π΄ 1 − π΅1 ππ΅1∗ − πΆ1 ππΆ1∗ , (9) π = π΄†1 πΆ1 + πΏ π΄ 1 π·1 π΅1† + πΏ π΄ 1 ππ π΅1 , ∗ ∗ 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 π΄ 1 π = πΆ1 , π΄ 2 ππ΅2 = πΆ2 . 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 π and π are Hermitian solutions of π΄ 2 π = πΆ2 , (11) ππ΅2 = π·2 , (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 π΄ 1 π = πΆ1 , ππ΅1 = π·1 , (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) π΄ 1 π·1 = πΆ1 π΅1 . π (π) = π΄ 3 − π΅3 π − (π΅3 π) , π (π, π) = π΄ 3 − π΅3 π − (π΅3 π) − πΆ3 ππ·3 − (πΆ3 ππ·3 ) . (10) (14) (16) where π is arbitrary. (b) Let π΄ 2 and πΆ2 be given. Then the following statements are equivalent: (1) equation (11) is consistent, (2) π π΄ 2 πΆ2 = 0, (17) π [π΄ 2 πΆ2 ] = π (π΄ 2 ) . (18) (3) In this case, the general solution can be written as π = 𴆠πΆ + πΏ π΄π, (19) where π is arbitrary. Lemma 2 ([22, Lemma 1.5, Theorem 2.3]). Let π΄ ∈ Cπ×π , β , and denote that π΅ ∈ Cπ×π , and π· ∈ Cπ×π β π΄ π΅ π = [ ∗ ], π΅ 0 π=[ πΏ=[ πΊ=[ 0 π ], π∗ 0 π΄ π΅ ], π΅∗ π· π ππΏ π ]. πΏ ππ∗ 0 (20) Journal of Applied Mathematics 3 (h) any π ∈ π» satisfies π ≥ 0 (π ≤ 0) if and only if maxπ∈π»π− (π) = 0 (maxπ∈π»π+ (π) = 0). Then one has the following (a) the following equalities hold π± (π) = π (π΅) + π± (π π΅ π΄π π΅ ) , (21) π± (π) = π (π) , (22) Lemma 6 (see [16]). Let π(π, π) = π΄ − π΅π − (π΅π)∗ − πΆππ· − (πΆππ·)∗ , where π΄, π΅, πΆ, and π· are given with appropriate sizes, and denote that π΄ π΅ πΆ π1 = [ π΅∗ 0 0 ] , ∗ [πΆ 0 0 ] (b) if R(π΅) ⊆ R(π΄), then π± (πΏ) = π± (π΄) + π± (π· − π΅∗ 𴆠π΅)). Thus π± (πΏ) = π± (π΄) if and only if R(π΅) ⊆ R(π΄) and π± (π· − π΅∗ 𴆠π΅) = 0, π΄ π΅ π·∗ π2 = [π΅∗ 0 0 ] , [π· 0 0 ] (c) π π 0 π± (πΊ) = [π∗ 0 π∗ ] − π (π) . [ 0 π 0 ] (23) π3 = [ (a) π[π΄ π΅] = π(π΄) + π(πΈπ΄ π΅) = π(π΅) + π(πΈπ΅ π΄), (d) (e) π [ πΈπΆπ΅ π΄ ] (f) π [ πΈπ΄πΈ πΆ π΅πΉ0π· ] (b) (c) = πΆ 0] π[π΄ π΅ = − π(π΅), π΄ π΅ 0 π[πΆ 0 πΈ] 0 π· 0 π΄ π΅ πΆ π·∗ [ π5 = π΅∗ 0 0 0 ] . [π· 0 0 0 ] Then one has the following: (1) the maximal rank of π(π, π) is − π(π·) − π(πΈ), , π΅ ∈ Cπ×π , πΆ ∈ Cπ×π Lemma 4 (see [15]). Let π΄ ∈ Cπ×π β β ,π ∈ Cπ×π , and π ∈ Cπ×π be given, and π ∈ Cπ×π be nonsingular. Then one has the following max π [π (π, π)] = min {π, π (π1 ) , π (π2 ) , π (π3 )} , π∈Cπ×π ,π (25) (2) the minimal rank of π(π, π) is ∗ (1) π± (ππ΄π ) = π± (π΄), min π [π (π, π)] π∈Cπ×π ,π (2) π± [ π΄0 πΆ0 ] = π± (π΄) + π± (πΆ), (3) π± [ π0∗ π΄ π 0] (4) π± [ πΏ π π΅∗ (26) = 2π (π3 ) − 2π (π΅) = π(π), π΅πΏ π 0 ] (24) π΄ π΅ πΆ π·∗ [ π4 = π΅∗ 0 0 0 ] , ∗ [πΆ 0 0 0 ] Lemma 3 (see [23]). Let π΄ ∈ Cπ×π , π΅ ∈ Cπ×π , and πΆ ∈ Cπ×π . Then they satisfy the following rank equalities: π[π΄ πΆ ] = π(π΄) + π(πΆπΉπ΄ ) = π(πΆ) + π(π΄πΉπΆ ), π΄ π [ πΆ π΅0 ] = π(π΅) + π(πΆ) + π(πΈπ΅ π΄πΉπΆ), π[π΅ π΄πΉπΆ] = π [ π΅0 π΄ πΆ ] − π(πΆ), π΄ π΅ πΆ π·∗ ], π΅∗ 0 0 0 π΄ π΅ 0 0 π∗ 0 π 0 + π(π) = π± [ π΅∗ + max {π’+ + π’− , V+ + V− , π’+ + V− , π’− + V+ } , ]. Lemma 5 (see [22, Lemma 1.4]). Let π be a set consisting of (square) matrices over Cπ×π , and let π» be a set consisting of . Then Then one has the following (square) matrices over Cπ×π β (a) π has a nonsingular matrix if and only if maxπ∈π π(π) = π; (b) any π ∈ π is nonsingular if and only if minπ∈π π(π) = π; (3) the maximal inertia of π(π, π) is max π± [π (π, π)] = min {π± (π1 ) , π± (π2 )} , π∈Cπ×π ,π (4) the minimal inertias of π(π, π) is min π± [π (π, π)] = π (π3 ) − π (π΅) π∈Cπ×π ,π (c) {0} ∈ π if and only if minπ∈π π(π) = 0; + max {π± (π1 ) − π (π4 ) , (d) π = {0} if and only if maxπ∈π π(π) = 0; (e) π» has a matrix π > 0 (π < 0) if and only if maxπ∈π»π+ (π) = π(maxπ∈π»π− (π) = π); (f) any π ∈ π» satisfies π > 0 (π < 0) if and only if minπ∈π»π+ (π) = π (minπ∈π»π− (π) = π); (g) π» has a matrix π ≥ 0 (π ≤ 0) if and only if minπ∈π»π− (π) = 0 (minπ∈π»π+ (π) = 0); (27) (28) π± (π2 ) − π (π5 )} , where π’± = π± (π1 ) − π (π4 ) , V± = π± (π2 ) − π (π5 ) . (29) Now we present the main theorem of this section. 4 Journal of Applied Mathematics Theorem 7. Let π΄ 1 ∈ Cπ×π ,πΆ1 ∈ Cπ×π , π΅1 ∈ Cπ×π , π·1 ∈ Cπ×π , π×π π΄ 2 ∈ Cπ‘×π , πΆ2 ∈ Cπ‘×π ,π΄ 3 ∈ Cβ , π΅3 ∈ Cπ×π , πΆ3 ∈ Cπ×π , π×π and π·3 ∈ 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 π and (11) by πΊ. Put π΄3 [ πΆ∗ [ 3 πΈ1 = [ 3 [πΆ1 π· [ π΅3∗ [ πΆ2 πΆ3 π·3∗ πΆ1∗ 0 π΄∗1 π΄1 0 0 0 0 0 π΅3 πΆ2∗ 0 0] ] 0 0] ], 0 π΄∗2 ] π΄2 0 ] π΄ 3 π·3∗ πΆ3 π·1 [ π·3 [ ∗ ∗ 0∗ π΅1 0 πΈ2 = π [ 3 π΅1 [π·1 πΆ [ π΅3∗ 0 0 πΆ 0 0 [ 2 π΄ 3 π΅3 [ π΅∗ [ ∗3 ∗ 0 πΈ3 = [ [π·1 πΆ3 0 [ πΆ1 π·3 0 [ πΆ2 π΄ 2 πΆ3∗ 0 0 0 π΅1∗ π΄1 0 0 0 π΅3 0 0 0 0 π΄2 πΆ3 0 0 0 0 0 π΅1∗ π΄1 0 0 0 πΆ2∗ π΄∗2 π΄ 3 π΅3 [ π΅∗ 0 [ 3 [ π·3 0 πΈ5 = [ [π·1∗ πΆ3∗ 0 [ [ 0 0 π΄ πΆ 2 2 [ π·3∗ πΆ2∗ π΄∗2 π΄3 [ π΅∗ [ 3∗ [ πΆ 3 πΈ4 = [ [ 0 [ [πΆ1 π·3 [ πΆ2 π·3∗ πΆ3 0 0 0 0 0 π΄∗1 π΄1 0 0 0 min π± [π (π, π)] = π (πΈ3 ) − π [ 0 π΄∗1 0 0 0 (34) π± (πΈ2 ) − π (πΈ5 )} . (30) Proof. By Lemma 1, the general solutions to (13) and (11) can be written as π = π΄†2 πΆ2 + πΏ π΄ 2 π, ] ] ] ], ] ] ] π = π΄†1 πΆ1 + πΏ π΄ 1 π·1 π΅1† + πΏ π΄ 1 ππ π΅1 , ] π = π΅3 πΏ π΄ 2 , πΆ3 π·1 0 ] ] π΅1 ] ]. 0 ] ] 0 ] 0 ] π = πΆ3 πΏ π΄ 1 , π½ = π π΅1 π·3 , π = π΄ 3 − π΅3 π΄†2 πΆ2 − (π΅3 π΄†2 πΆ2 ) ∗ (36) − πΆ3 (π΄†1 πΆ1 + πΏ π΄ 1 π·1 π΅1† ) π·3 ∗ − (πΆ3 (π΄†1 πΆ1 + πΏ π΄ 1 π·1 π΅1† ) π·3 ) . (a) the maximal rank of (10) subject to (13) and (11) is max π [π (π, π)] Substituting (36) into (10) yields π (π, π) = π − ππ − (ππ)∗ − πππ½ − (πππ½)∗ . π∈πΊ,π∈π = min {π, π (πΈ1 ) − 2π (π΄ 1 ) − 2π (π΄ 2 ) , (35) where π and π are arbitrary matrices with appropriate sizes. Put Then one has the following: (31) π (πΈ2 ) − 2π (π΅1 ) − 2π (π΄ 2 ) , (37) Clearly π is Hermitian. It follows from Lemma 6 that max π [π (π, π)] π (πΈ3 ) − 2π (π΄ 2 ) − π (π΄ 1 ) − π (π΅1 )} , π∈πΊ,π∈π = max π (π − ππ − (ππ)∗ − πππ½ − (πππ½)∗ ) (b) the minimal rank of (10) subject to (13) and (11) is π,π min π [π (π, π)] π∈πΊ,π∈π = 2π (πΈ3 ) − 2π [ π΅3 ] π΄2 + max {π± (πΈ1 ) − π (πΈ4 ) , ] π·3∗ πΆ1∗ 0 0 0 0 0 0 0 0 ], ] ] π± (πΈ2 ) − π (π΅1 ) − π (π΄ 2 )} , (33) π∈πΊ,π∈π πΆ2∗ π΄∗2 ] ] 0 0 0 max π± [π (π, π)] = min {π± (πΈ1 ) − π (π΄ 1 ) − π (π΄ 2 ) , π∈πΊ,π∈ππ₯ (d) the minimal inertia of (10) subject to (13) and (11) is π΅3 πΆ2∗ 0 0] ] 0 0] , ∗] 0 π΄ 2] π΄2 0 ] π·3∗ (c) the maximal inertia of (10) subject to (13) and (11) is (38) = min {π, π (π1 ) , π (π2 ) , π (π3 )} , π΅3 ] π΄2 + max {π (πΈ1 ) − 2π (πΈ4 ) , π (πΈ2 ) − 2π (πΈ5 ) , π+ (πΈ1 ) + π− (πΈ2 ) − π (πΈ4 ) − π (πΈ5 ) , π− (πΈ1 ) + π+ (πΈ2 ) − π (πΈ4 ) − π (πΈ5 )} , min π [π (π, π)] π∈πΊ,π∈π (32) = max π (π − ππ − (ππ)∗ − πππ½ − (πππ½)∗ ) π,π = 2π (π3 ) − 2π (π) + max {π + + π − , π‘+ + π‘− , π + + π‘− , π − + π‘+ } , (39) Journal of Applied Mathematics 5 max π± [π (π, π)] By πΆ1 π΅1 = π΄ 1 π·1 , we obtain π∈πΊ,π∈π = max π (π − ππ − (ππ)∗ − πππ½ − (πππ½)∗ ) π,π (40) π π π½∗ ] [ π (π2 ) = π [π∗ 0 0 ] = min {π± (π1 ) , π± (π2 )} , [π½ min π± [π (π, π)] 0 0] π·3∗ πΆ3 π·1 π΅3 πΆ2∗ π΄3 π∈πΊ,π∈π = max π (π − ππ − (ππ)∗ − πππ½ − (πππ½)∗ ) π,π (41) = π (π3 ) − π (π) + max {π ± , π‘± } , [ [ π·3 0 [ [π·∗ πΆ∗ π΅∗ = π[ 1 3 1 [ [ π΅∗ 0 [ 3 0 [ πΆ2 where π΅1 0 0 0 ] 0] ] 0 0] ] ] ∗] 0 π΄ 2] π΄2 0 ] 0 − 2π (π΅1 ) − 2π (π΄ 2 ) , π π π π1 = [π∗ 0 0 ] , ∗ [π 0 0 ] π (π3 ) = π [ π π π½ π2 = [π∗ 0 0 ] , [ π½ 0 0] π π π π½∗ π3 = [ ∗ ], π 0 0 0 (42) ∗ [π Now, we simplify the ranks and inertias of block matrices in (38)–(41). By Lemma 4, block Gaussian elimination, and noting that ∗ ∗ † = (πΌ − π π) = πΌ − π (π ) = π π∗ , (43) π π π π (π1 ) = π [π∗ 0 0 ] ∗ [π 0 0 ] πΆ3 π·3∗ πΆ1∗ π΅3 πΆ2∗ 0 π΄∗1 0 0] ] π΄1 0 0 0] ] 0 0 0 π΄∗2 ] 0 0 π΄2 0 ] − 2π (π΄ 1 ) − 2π (π΄ 2 ) . 0 0 0 0 π΄1 π΄2 0 ] 0 π΄∗2 ] ] π΅1∗ 0 ] ] ] 0 0] ] 0 0] 0 0 0] π΄3 [ π΅∗ [ 3∗ [ πΆ 3 = π[ [ 0 [ [πΆ1 π·3 [ πΆ2 π΅3 0 0 0 0 π΄2 πΆ3 π·3∗ πΆ2∗ π·3∗ πΆ1∗ 0 0 π΄∗2 0 ] ] 0 0 0 π΄∗1 ] ] ∗ 0 π΅1 0 0 ] ] π΄1 0 0 0 ] 0 0 0 0 ] − π (π΅1 ) − 2π (π΄ 2 ) − 2π (π΄ 1 ) , π π π π½∗ π (π5 ) = π [π∗ 0 0 0 ] ∗ [π 0 0 0 ] π΄3 [ π΅∗ [ 3 [ π·3 = π[ [π·1∗ πΆ3∗ [ [ 0 [ πΆ2 we have the following: π΄3 [ πΆ∗ [ 3 = π[ 3 [πΆ1 π· [ π΅3∗ [ πΆ2 0 [ ] π (π4 ) = π [π∗ 0 0 0 ] π‘± = π± (π2 ) − π (π5 ) . ∗ π΅3 πΆ3∗ π·3∗ πΆ2∗ π π π π½∗ π π π π½∗ [ π5 = π∗ 0 0 0 ] , [ π½ 0 0 0] † [ ∗ [ π΅3 [ [ = π [π·1∗ πΆ3∗ [ [πΆ π· [ 1 3 [ πΆ2 ] − π (π΅1 ) − 2π (π΄ 2 ) − π (π΄ 1 ) , π π π π½∗ π4 = [π∗ 0 0 0 ] , ∗ [π 0 0 0 ] πΏ∗π π∗ 0 0 0 π΄3 ∗ π ± = π± (π1 ) − π (π4 ) , π π π π½∗ π΅3 0 0 0 0 π΄2 πΆ3 π·3∗ 0 0 0 0 0 π΄∗1 π΄1 0 0 0 πΆ2∗ πΆ3 π·1 π΄∗2 0 ] ] 0 π΅1 ] ] 0 0 ] ] 0 0 ] 0 0 ] − 2π (π΅1 ) − 2π (π΄ 2 ) − π (π΄ 1 ) . (44) (45) By Lemma 2, we can get the following: π π π π± (π1 ) = π± [π∗ 0 0 ] ∗ [π 0 0 ] 6 Journal of Applied Mathematics π΄3 [ πΆ∗ [ 3 = π± [ 3 [πΆ1 π· [ π΅3∗ [ πΆ2 πΆ3 π·3∗ πΆ1∗ 0 π΄∗1 π΄1 0 0 0 0 0 π΅3 πΆ2∗ 0 0] ] 0 0] ] 0 π΄∗2 ] π΄2 0 ] (e) π΄ 3 − π΅3 π − (π΅3 π)∗ − πΆ3 ππ·3 − (πΆ3 ππ·3 )∗ > 0 for all π ∈ πΊ and π ∈ π if and only if π (πΈ3 ) − π [ − π (π΄ 1 ) − π (π΄ 2 ) , (46) π π π½∗ π± (π2 ) = π± [π∗ 0 0 ] [ π½ 0 0] π΄ 3 π·3∗ πΆ3 π·1 [ π·3 [ ∗ ∗ 0∗ π΅1 0 = π± [ 3 π΅1 [π·1 πΆ [ π΅3∗ 0 0 πΆ 0 0 2 [ π΅3 πΆ2∗ 0 0] ] 0 0] ] 0 π΄∗2 ] π΄2 0 ] (47) Corollary 8. Let π΄ 1 , πΆ1 , π΅1 , π·1 , π΄ 2 , πΆ2 , π΄ 3 , π΅3 , πΆ3 , π·3 , and πΈπ , (π = 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 π and (11) by πΊ. Then, one has the following: (a) there exist π ∈ πΊ and π ∈ π such that π΄ 3 − π΅3 π − (π΅3 π)∗ − πΆ3 ππ·3 − (πΆ3 ππ·3 )∗ > 0 if and only if (48) (b) there exist π ∈ πΊ and π ∈ π such that π΄ 3 − π΅3 π − (π΅3 π)∗ − πΆ3 ππ·3 − (πΆ3 ππ·3 )∗ < 0 if and only if π− (πΈ1 ) − π (π΄ 1 ) − π (π΄ 2 ) ≥ π, π− (πΈ2 ) − π (π΅1 ) − π (π΄ 2 ) ≥ π, (49) (c) there exist π ∈ πΊ and π ∈ π such that π΄ 3 − π΅3 π − (π΅3 π)∗ − πΆ3 ππ·3 − (πΆ3 ππ·3 )∗ ≥ 0 if and only if π (πΈ3 ) − π [ π (πΈ3 ) − π [ π΅3 ] + π− (πΈ1 ) − π (πΈ4 ) ≤ 0, π΄2 π΅3 ] + π− (πΈ2 ) − π (πΈ5 ) ≤ 0. π΄2 π΅3 ] + π+ (πΈ1 ) − π (πΈ4 ) ≤ 0, π΄2 π΅ π (πΈ3 ) − π [ 3 ] + π+ (πΈ2 ) − π (πΈ5 ) ≤ 0, π΄2 π΅3 ] + π− (πΈ1 ) − π (πΈ4 ) = π π΄2 (52) (50) (51) (53) (g) π΄ 3 − π΅3 π − (π΅3 π)∗ − πΆ3 ππ·3 − (πΆ3 ππ·3 )∗ ≥ 0 for all π ∈ πΊ and π ∈ π if and only if π− (πΈ1 ) − π (π΄ 1 ) − π (π΄ 2 ) ≤ 0 ππ π− (πΈ2 ) − π (π΅1 ) − π (π΄ 2 ) ≤ 0, (54) (h) π΄ 3 − π΅3 π − (π΅3 π)∗ − πΆ3 ππ·3 − (πΆ3 ππ·3 )∗ ≤ 0 for all π ∈ πΊ and π ∈ π if and only if π+ (πΈ1 ) − π (π΄ 1 ) − π (π΄ 2 ) ≤ 0 ππ π+ (πΈ2 ) − π (π΅1 ) − π (π΄ 2 ) ≤ 0, (55) (i) there exist π ∈ πΊ and π ∈ π such that π΄ 3 − π΅3 π − (π΅3 π)∗ −πΆ3 ππ·3 −(πΆ3 ππ·3 )∗ is nonsingular if and only if π (πΈ1 ) − 2π (π΄ 1 ) − 2π (π΄ 2 ) ≥ π, π (πΈ2 ) − 2π (π΅1 ) − 2π (π΄ 2 ) ≥ π, (d) there exist π ∈ πΊ and π ∈ π such that π΄ 3 − π΅3 π − (π΅3 π)∗ − πΆ3 ππ·3 − (πΆ3 ππ·3 )∗ ≤ 0 if and only if π (πΈ3 ) − π [ π (πΈ3 ) − π [ π΅ ππ π (πΈ3 ) − π [ 3 ] + π− (πΈ2 ) − π (πΈ5 ) = π, π΄2 Substituting (44)-(47) into (38) and (41) yields (31)–(34), respectively. π+ (πΈ2 ) − π (π΅1 ) − π (π΄ 2 ) ≥ π. π΅ ππ π (πΈ3 ) − π [ 3 ] + π+ (πΈ2 ) − π (πΈ5 ) = π, π΄2 (f) π΄ 3 − π΅3 π − (π΅3 π)∗ − πΆ3 ππ·3 − (πΆ3 ππ·3 )∗ < 0 for all π ∈ πΊ and π ∈ π if and only if − π (π΅1 ) − π (π΄ 2 ) . π+ (πΈ1 ) − π (π΄ 1 ) − π (π΄ 2 ) ≥ π, π΅3 ] + π+ (πΈ1 ) − π (πΈ4 ) = π π΄2 (56) π (πΈ3 ) − 2π (π΄ 2 ) − π (π΄ 1 ) − π (π΅1 ) ≥ π. 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 π΄ 1 ∈ Cπ×π , πΆ1 ∈ Cπ×π , π΅1 ∈ Cπ×π , π·1 ∈ Cπ×π , π΄ 2 ∈ Cπ‘×π , and πΆ2 ∈ Cπ‘×π , 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 π and (11) by πΊ. Put 0 [πΌ [ π»1 = [ [πΆ1 [−πΌ [πΆ2 πΌ 0 π΄1 0 0 πΆ1∗ π΄∗1 = π (πΈ3 ) − π + max {π± (π»1 ) − π (π»4 ) , π± (π»2 ) − π (π»5 )} . (58) πΆ2∗ −πΌ 0 0] ] 0 0 0] ], 0 0 π΄∗2 ] 0 π΄2 0 ] Proof. By letting π΄ 3 = 0, π΅3 = −πΌ, πΆ3 = πΌ, and π·3 = πΌ in Theorem 7, we can get the results. Corollary 10. Let π΄ 1 , πΆ1 , π΅1 , π·1 , π΄ 2 , πΆ2 , and π»π , (π = 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 π and (11) by πΊ. Then, one has the following: πΆ2∗ 0 [ πΌ [ ∗ π»2 = π [ [π·1 [ −πΌ [ πΆ2 πΌ 0 π΅1∗ 0 0 π·1 π΅1 0 0 0 −πΌ 0 0] ] 0 0] ], 0 π΄∗2 ] π΄2 0 ] 0 [ −πΌ [ ∗ π»3 = [ [π·1 [ πΆ1 [ πΆ2 −πΌ 0 0 0 π΄2 πΌ 0 0 π΄1 0 πΌ πΆ2∗ 0 π΄∗2 ] ] π΅1∗ 0 ] ], 0 0] 0 0] πΌ 0 0 π΅1∗ 0 0 πΆ2∗ π΄∗2 0 [−πΌ [ [πΌ π»4 = [ [0 [ [πΆ1 [πΆ2 −πΌ 0 0 0 0 π΄2 πΌ 0 0 0 π΄1 0 0 [ −πΌ [ [ πΌ π»5 = [ [π·1∗ [ [0 [ πΆ2 −πΌ 0 0 0 0 π΄2 πΌ πΌ πΆ2∗ 0 0 π΄∗2 0 0 0 0 π΄∗1 0 π΄1 0 0 0 0 0 (a) there exist π ∈ πΊ and π ∈ π such that (π + π∗ ) > (π + π∗ ) if and only if π+ (π»1 ) − π (π΄ 1 ) − π (π΄ 2 ) ≥ π, (57) πΆ1∗ 0] ] 0 π΄∗1 ] ], 0 0] ] 0 0] 0 0] π·1 0] ] π΅1 ] ]. 0] ] 0] 0] Then one has the following: max π [(π + π∗ ) − (π + π∗ )] π∈πΊ,π∈π = min {π, π (π»1 ) − 2π (π΄ 1 ) − 2π (π΄ 2 ) , π (π»2 ) − 2π (π΅1 ) −2π (π΄ 2 ) , π (π»3 ) − 2π (π΄ 2 ) − π (π΄ 1 ) − π (π΅1 )} , min π [(π + π∗ ) − (π + π∗ )] π∈πΊ,π∈π = 2π (π»3 ) − 2π + max {π (π»1 ) − 2π (π»4 ) , π (π»2 ) − 2π (π»5 ) , π+ (π»1 ) + π− (π»2 ) − π (π»4 ) − π (π»5 ) , π− (π»1 ) + π+ (π»2 ) − π (π»4 ) − π (π»5 )} , max π± [(π + π∗ ) − (π + π∗ )] π∈πΊ,π∈π = min {π± (π»1 ) − π (π΄ 1 ) − π (π΄ 2 ) , π± (π»2 ) − π (π΅1 ) − π (π΄ 2 )} , min π± [(π + π∗ ) − (π + π∗ )] π∈πΊ,π∈π π+ (π»2 ) − π (π΅1 ) − π (π΄ 2 ) ≥ π. (59) (b) there exist π ∈ πΊ and π ∈ π such that (π + π∗ ) < (π + π∗ ) if and only if π− (π»1 ) − π (π΄ 1 ) − π (π΄ 2 ) ≥ π, π− (π»2 ) − π (π΅1 ) − π (π΄ 2 ) ≥ π, (60) (c) there exist π ∈ πΊ and π ∈ π such that (π + π∗ ) ≥ (π + π∗ ) if and only if π (π»3 ) − π + π− (π»1 ) − π (π»4 ) ≤ 0, π (π»3 ) − π + π− (π»1 ) − π (π»4 ) ≤ 0, (61) (d) there exist π ∈ πΊ and π ∈ π such that (π + π∗ ) ≤ (π + π∗ ) if and only if π (π»3 ) − π + π+ (π»1 ) − π (π»4 ) ≤ 0, π (π»3 ) − π + π+ (π»2 ) − π (π»5 ) ≤ 0, (62) (e) (π + π∗ ) > (π + π∗ ) for all π ∈ πΊ and π ∈ π if and only if π (π»3 ) − π + π+ (π»1 ) − π (π»4 ) = π ππ π (π»3 ) − π + π+ (π»2 ) − π (π»5 ) = π, (63) (f) (π + π∗ ) < (π + π∗ ) for all π ∈ πΊ and π ∈ π if and only if π (π»3 ) − π + π− (π»1 ) − π (π»4 ) = π ππ π (π»3 ) − π + π− (π»2 ) − π (π»5 ) = π, (64) (g) (π + π∗ ) ≥ (π + π∗ ) for all π ∈ πΊ and π ∈ π if and only if π− (π»1 ) − π (π΄ 1 ) − π (π΄ 2 ) ≤ 0 ππ π− (π»2 ) − π (π΅1 ) − π (π΄ 2 ) ≤ 0, (65) 8 Journal of Applied Mathematics (h) (π + π∗ ) ≤ (π + π∗ ) for all π ∈ πΊ and π ∈ π if and only if π+ (π»1 ) − π (π΄ 1 ) − π (π΄ 2 ) ≤ 0 (66) ππ π+ (π»2 ) − π (π΅1 ) − π (π΄ 2 ) ≤ 0, ∗ (i) there exist π ∈ πΊ and π ∈ π such that (π + π ) − (π + π∗ ) is nonsingular if and only if where π1 , π2 , π1 , π2 , π1 , and π2 are arbitrary matrices over C with appropriate sizes. Now we give the main theorem of this section. Theorem 12. Let π΄ π , πΆπ , (π = 1, 2, 3), π΅π , and π·π , (π = 1, 3) be given. Set π΄ = π΅3 πΏ π΄ 2 , π (π»1 ) − 2π (π΄ 1 ) − 2π (π΄ 2 ) ≥ π, π (π»2 ) − 2π (π΅1 ) − 2π (π΄ 2 ) ≥ π, πΆ = π π΅1 π·3 , (67) πΊ = πΆπ π΄ , π (π»3 ) − 2π (π΄ 2 ) − π (π΄ 1 ) − π (π΅1 ) ≥ π. π = πΉ∗ πΏ πΊ, 4. The Solvability Conditions and the General Solution to System (6) (1) equation (5) is consistent, (2) π ππ π΄πΈ = 0, πΏ π΅ πΈπΏ π΅ = 0, − πΆ3 (π΄†1 πΆ1 + πΏ π΄ 1 π·1 π΅1† ) π·3 (69) πΈ1 πΆ1∗ π΄ 1 π [π΄∗1 0 0 ] = 2π [πΆ1∗ π΄ 1 ] . [ πΆ1 0 0 ] In this case, the general solution of (5) can be expressed as ∗ 1 π = [𴆠πΈπ΅† − 𴆠π΅∗ π† πΈπ΅† − 𴆠π(𡆠) πΈπ† π΄∗ 𡆠2 ∗ (72) ∗ − π·3∗ (π΄†1 πΆ1 + πΏ π΄ 1 π·1 π΅1† ) πΆ3∗ , πΈ = π π΄ π·π π΄ . (73) Then the following statements are equivalent: (1) system (6) is consistent, (2) the equalities in (14) and (17) hold, and π πΉ πΈπ πΉ = 0, πΏ πΊπΈπΏ πΊ = 0, (74) (3) the equalities in (15) and (18) hold, and πΈ1 π΅1 πΆ1∗ π΄ 1 ] = π [π΅1 πΆ1∗ π΄ 1 ] + π (π΄ 1 ) , π΄∗1 0 0 0 πΈ1 π΅1 π΄ 1 π [π΄∗1 0 0 ] = 2π [π΅1 π΄ 1 ] , ∗ [ π΅1 0 0 ] π = πΊ∗ πΏ π, (68) (3) π[ (71) π = π πΉ πΊ∗ , ∗ π ππ πΉ πΈ = 0, π π΄ πΈπ π΄ = 0, πΉ = π π΄ π΅, π· = π΄ 3 − π΅3 π΄†2 πΆ2 − (π΅3 π΄†2 πΆ2 ) 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 π΄ 1 ∈ Cπ×π1 , π΅1 ∈ Cπ×π2 , πΆ1 ∈ be given. Let π΄ = π π΄ 1 π΅1 , π΅ = πΆ1 π π΄ 1 , Cπ×π , and πΈ1 ∈ Cπ×π β πΈ = π π΄ 1 πΈ1 π π΄ 1 , π = π π΄ π΅∗ , π = π΄∗ πΏ π΅ , and π = π΅∗ πΏ π. Then the following statements are equivalent: π΅ = πΆ3 πΏ π΄ 1 , ∗ +𴆠πΈ(π† ) + (π† ) πΈπ΅† π† π] + πΏ π΄π1 + π2 π π΅ + π1 πΏ π πΏ π + π ππ2∗ πΏ π − 𴆠ππ2 π ππ΄∗ 𡆠, π΄3 [ π΅∗ [ 3 π[ [ πΆ1∗π·3∗ [π·1 πΆ3 [ πΆ2 πΆ3 π·3∗ 0 0 π΄1 0 0 π΅1∗ 0 0 π΅3 πΆ2∗ πΆ3 π·3∗ 0 π΄∗2 ] ] [π΄ 1 0 [ 0 0] ] = π [ 0 π΅1∗ 0 0] [0 0 π΄2 0 ] π΅3 0] ] + π [π΄ 2 ] , 0] π΅3 π΄ 2] πΆ3 0 0 π΄1 0 π΅3 π·3∗ πΆ1∗ πΆ2∗ 0 π΄∗1 0] πΆ3 π΅3 ] [π΄ 1 0 ] , 0 0 π΄∗2 ] = 2π ] 0 0 0] [ 0 π΄ 2] π΄2 0 0] π΄ 3 π·3∗ [ π·3 0 [ ∗ π[ [ π΅∗ 3 ∗ 0∗ [π·1 πΆ3 π΅1 0 [ πΆ2 π΅3 πΆ3 π·1 πΆ2∗ 0 π΅1 0] π·3∗ π΅3 ] ∗ ] [ 0 0 π΄∗2 ] ] = 2π π΅1 0 . 0 0 0] 0 π΄ 2] [ π΄2 0 0] π΄3 [ πΆ∗ [ ∗3 π[ [ π΅3 [πΆ1 π·3 [ πΆ2 (75) ∗ π = π΄†1 [πΈ1 − π΅1 ππΆ1 − (π΅1 ππΆ1 ) ] In this case, the general solution of system (6) can be expressed as 1 ∗ − π΄†1 [πΈ1 − π΅1 ππΆ1 − (π΅1 ππΆ1 ) ] π΄ 1 π΄†1 2 − π΄†1 π1 π΄∗1 + π1∗ π΄ 1 π΄†1 + πΏ π΄ 1 π2 , π = π΄†2 πΆ2 + πΏ π΄ 2 π, (70) π = π΄†1 πΆ1 + πΏ π΄ 1 π·1 π΅1† + πΏ π΄ 1 ππ π΅1 , (76) Journal of Applied Mathematics 9 π·3∗ π΅3 πΆ3 π·1 πΆ2∗ π΄3 where ∗ 1 π = [πΉ† πΈπΊ† − πΉ† πΊ∗ π† πΈπΊ† − πΉ† π(πΊ† ) πΈπ† πΉ∗ πΊ† 2 ∗ ∗ +πΉ† πΈ(π† ) + (π† ) πΈπΊ† π† π] + πΏ πΉ π1 + π2 π πΊ + π1 πΏ π πΏ π + π ππ2∗ πΏ π − πΉ† ππ2 π ππΉ∗ πΊ† , † ∗ [ [ π·3 0 0 [ [ π΅∗ 0 0 πΏ πΊπΈπΏ πΊ = 0 ⇐⇒ π [ 3 [ ∗ ∗ [π· πΆ π΅∗ 0 [ 1 3 1 0 π΄2 [ πΆ2 1 − 𴆠[π· − π΅ππΆ − (π΅ππΆ)∗ ] π΄π΄† 2 † ∗ − π΄ π1 π΄ + π1∗ π΄π΄† + πΏ π΄π2 , (77) (1) ⇔ (2): We separate the four equations in system (6) into three groups: π΄ 1 π = πΆ1 , = π [π΅ πΆ∗ π΄] + π (π΄) πΆ3 π·3∗ [π΄ 1 0 = π[ [ 0 π΅1∗ [0 0 π΄ 2 π = πΆ2 , πΆ3 π·3∗ 0 0 π΄1 0 0 π΅1∗ 0 0 πΆ3 π΅3 = 2π [π΄ 1 0 ] , [ 0 π΄ 2] ∗ π ππ πΉ πΈ = 0, (78) π΅3 πΆ2∗ 0 π΄∗2 ] ] 0 0] ] 0 0] π΄2 0 ] ∗ π΅3 0 0 0 π΄2 (82) (83) π πΉ πΈπ πΉ = 0, πΏ πΊπΈπΏ πΊ = 0. (84) We know by Lemma 11 that the general solution of (83) can be expressed as (77). In Theorem 12, let π΄ 1 and π·1 vanish. Then we can obtain the general solution to the following system: π΄ 2 π = πΆ2 , ππ΅1 = π·1 , ∗ ∗ π΅3 π + (π΅3 π) + πΆ3 ππ·3 + (πΆ3 ππ·3 ) = π΄ 3 . (85) Corollary 13. Let π΄ 2 , πΆ2 , π΅1 , π·1 , π΅3 , πΆ3 , π·3 , and π΄ 3 = π΄∗3 be given. Set π΄ = π΅3 πΏ π΄ 2 , πΆ = π π΅1 π·3 , πΉ = π π΄ πΆ3 , By a similar approach, we can obtain that πΆ3 0 0 π΄1 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 π΅3 0] ] + π [π΄ 2 ] . 0] π΅3 π΄ 2] π΄3 [ πΆ∗ [ ∗3 π πΉ πΈπ πΉ = 0 ⇐⇒ π [ [ π΅3 [πΆ1 π·3 [ πΆ2 (80) π΄π + (π΄π)∗ + π΅ππΆ + (π΅ππΆ)∗ = π·. π΅3 0 0 π΄∗2 ] ] 0 0] ] 0 0] π΄2 0 ] π΅3 0] ] + π [π΄ 2 ] 0] π΅3 π΄ 2] π΄3 [ π΅∗ [ 3 ⇐⇒ π [ [ πΆ1∗π·3∗ [π·1 πΆ3 [ πΆ2 ππ΅1 = π·1 , 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 π· π΅ πΆ∗ π΄ ] π΄∗ 0 0 0 πΆ3 π·3∗ [π΄ 1 0 = π[ [ 0 π΅1∗ [0 0 0] ]. π΅3 π + (π΅3 π) + πΆ3 ππ·3 + (πΆ3 ππ·3 ) = π΄ 3 . π ππ πΉ πΈ = 0 ⇐⇒ π (π ππ πΉ πΈ) πΆ3 π·3∗ 0 0 π΄1 0 0 π΅1∗ 0 0 0 (79) Proof. (2) ⇔ (3): Applying Lemma 3 and Lemma 11 gives π· [π΅∗ [ 3 ⇐⇒ π [ [0 [0 [0 0 [ 0 π΄ 2] where π1 , π2 , π1 , π2 , π1 , and π2 are arbitrary matrices over C with appropriate sizes. = 0 ⇐⇒ π [ 0 π·3∗ π΅3 [ = 2π [ π΅1∗ π = π΄ [π· − π΅ππΆ − (π΅ππΆ) ] ] 0] ] π΄∗2 ] ] ] 0] ] 0] π΅1 π·3∗ πΆ1∗ π΄∗1 0 0 0 πΆ2∗ 0] ] π΄∗2 ] ] 0] 0] πΊ = πΆπ π΄ , π = π πΉ πΊ∗ , π = πΉ∗ πΏ πΊ , π = πΊ∗ πΏ π, π· = π΄3 − π΅3 π΄†2 πΆ2 − (86) ∗ (π΅3 π΄†2 πΆ2 ) ∗ − πΆ3 π·1 π΅1† π·3 − (πΆ3 π·1 π΅1† π·3 ) , πΈ = π π΄ π·π π΄ . 10 Journal of Applied Mathematics Then the following statements are equivalent: (1) system (85) is consistent (2) π π΄ 2 πΆ2 = 0, π·1 πΏ π΅1 = 0, π πΉ πΈπ πΉ = 0, π ππ πΉ πΈ = 0, πΏ πΊπΈπΏ πΊ = 0, (87) (88) (3) π [π΄ 2 πΆ2 ] = π (π΄ 2 ) , π΄3 [ π΅∗ [ π [ ∗3 ∗ π·1 πΆ3 [ πΆ2 πΆ3 π·3∗ 0 0 0 π΅1∗ 0 0 π΄3 [πΆ∗ 3 π[ [ π΅3∗ [ πΆ2 [ π·1 ] = π (π΅1 ) , π΅1 π΅3 πΆ2∗ πΆ3 π·3∗ π΅3 0 π΄∗2 ] ] = π [ 0 π΅1∗ 0 ] + π [π΄ 2 ] , 0 0] π΅3 [ 0 0 π΄ 2] π΄2 0 ] πΆ3 π΅3 πΆ2∗ 0 0 0] ] = 2π [πΆ3 π΅3 ] , 0 0 π΄∗2 ] 0 π΄2 0 π΄2 0 ] π΄ 3 π·3∗ [ π·3 0 [ ∗ π΅ π[ 3 [ ∗ ∗ 0∗ [π·1 πΆ3 π΅1 0 [ πΆ2 π΅3 πΆ3 π·1 πΆ2∗ 0 π΅1 0] π·3∗ π΅3 ∗] ] [ 0 0 π΄ 2 ] = 2π π΅1∗ 0 ] . 0 0 0] [ 0 π΄ 2] π΄2 0 0] (89) In this case, the general solution of system (6) can be expressed as π = π΄†2 πΆ2 + πΏ π΄ 2 π, π = π·1 π΅1† + ππ π΅1 , (90) where π= ∗ 1 † † [πΉ πΈπΊ − πΉ† πΊ∗ π† πΈπΊ† − πΉ† π(πΊ† ) πΈπ† πΉ∗ πΊ† 2 ∗ ∗ +πΉ† πΈ(π† ) + (π† ) πΈπΊ† π† π] + πΏ πΉ π1 + π2 π πΊ + π1 πΏ π πΏ π + π ππ2∗ πΏ π − πΉ† ππ2 π ππΉ∗ πΊ† , ∗ π = 𴆠[π· − πΆ3 ππΆ − (πΆ3 ππΆ) ] 1 ∗ − 𴆠[π· − πΆ3 ππΆ − (πΆ3 ππΆ) ] π΄π΄† 2 − 𴆠π1 π΄∗ + π1∗ π΄π΄† + πΏ π΄π2 , (91) where π1 , π2 , π1 , π2 , π1 , and π2 are arbitrary matrices over C with appropriate sizes. 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