Equalities and inequalities for Hermitian solutions and Her- mitian definite solutions of the two matrix equations AX = B and AXA∗ = B YONGGE TIAN Abstract. This paper studies algebraic properties of Hermitian solutions and Hermitian definite solutionsofthetwotypesofmatrixequationAX =B andAXA∗ =B. Wefirstestablishavariety ofrankandinertia formulasforcalculatingthe maximalandminimalranksandinertiasofHermi- tian solutions and Hermitian definite solutions of the matrix equations AX =B and AXA∗ =B, and then use them to characterize many qualities and inequalities for Hermitian solutions and 3 Hermitian definite solutions of the two matrix equations and their variations. 1 0 2 Mathematics Subject Classifications. 15A03, 15A09, 15A24, 15B57. n Keywords. Matrixequation,Hermitiansolution,Hermitiandefinite solution,generalizedinverse, a rank, inertia, matrix equality, matrix inequality, Lo¨wner partial ordering. J 7 1 1 Introduction ] A Consider the following two well-known linear matrix equations R AX =B (1.1) . h t and a m AXA∗ =B, (1.2) [ both of which are simplest cases of various types of linear matrix equation (with symmetric pat- 1 terns), and are the starting point of many advanced study on complicated matrix equations. A v huge amount of results about the two equations and applications were given in the literature. In 8 particular,manyproblemsonalgebraicpropertiesofsolutionsofthetwomatrixequationswereex- 3 plicitlycharacterizedbyusingformulasforranksandinertiasofmatrices. Inthispaper,theauthor 2 focuses on Hermitian solutions or Hermitian definite solutions of (1.1) and (1.2), and studies the 4 . following optimization problems on the ranks and inertias of Hermitian solutions and Hermitian 1 definite solutions of (1.1) and (1.2): 0 3 1 Problem 1.1 Let A, B ∈ Cm×n be given, and assume that (1.1) has a Hermitian solution or : HermitiandefinitesolutionX ∈Cn×n. Inthiscase,establishformulasforcalculatingtheextremal v ranks and inertias of i X X −P, (1.3) r a where P ∈ Cn×n is a given Hermitian matrix, and use the formulas to characterize behaviors of these Hermitian solution and definite solution, in particular, to give necessary and sufficient conditions for the four inequalities X ≻P, X 0 i (λA)= ± , (1.21) ± i (A) if λ<0 ∓ (cid:26) A 0 i =i (A)+i (B), (1.22) ± 0 B ± ± (cid:20) (cid:21) 0 Q 0 Q i =i =r(Q). (1.23) + Q∗ 0 − Q∗ 0 (cid:20) (cid:21) (cid:20) (cid:21) Lemma 1.8 ([7]) Let A∈Cm×n, B ∈Cm×k, C ∈Cl×n and D ∈Cl×k. Then, r[A, B]=r(A)+r(E B)=r(B)+r(E A), (1.24) A B A r =r(A)+r(CF )=r(C)+r(AF ), (1.25) C A C (cid:20) (cid:21) A B r =r(B)+r(C)+r(E AF ). (1.26) C 0 B C (cid:20) (cid:21) 3 Lemma 1.9 ([10]) Let A∈Cm, B ∈Cm×n, D ∈Cn, and let H H A B A B M = , M = . (1.27) 1 B∗ 0 2 B∗ D (cid:20) (cid:21) (cid:20) (cid:21) Then, the following expansion formulas hold i (M )=r(B)+i (E AE ), r(M )=2r(B)+r(E AE ), (1.28) ± 1 ± B B 1 B B 0 E B 0 E B i (M )=i (A)+i A , r(M )=r(A)+r A . (1.29) ± 2 ± ± B∗E D−B∗A†B 2 B∗E D−B∗A†B A A (cid:20) (cid:21) (cid:20) (cid:21) Under the condition A<0, i (M )=r[A, B], i (M )=r(B), r(M )=r[A, B]+r(B). (1.30) + 1 − 1 1 Under the condition R(B)⊆R(A), i (M )=i (A)+i (D−B∗A†B), r(M )=r(A)+r(D−B∗A†B). (1.31) ± 2 ± ± 2 Some general rank and inertia expansion formulas derived from (1.24)–(1.29) are given below A B A B 0 r =r −r(P), (1.32) E C 0 C 0 P P (cid:20) (cid:21) (cid:20) (cid:21) A B A BF r Q =r C 0 −r(Q), (1.33) C 0   (cid:20) (cid:21) 0 Q  A B 0 A BF r Q =r C 0 P −r(P)−r(Q), (1.34) (cid:20)EPC 0 (cid:21)  0 Q 0   A B 0 A BF i P =i B∗ 0 P∗ −r(P), (1.35) ±(cid:20)FPB∗ 0 (cid:21) ± 0 P 0   A B Q E AE E B i Q Q Q =i B∗ D 0 −r(Q). (1.36) ±(cid:20) B∗EQ D (cid:21) ±Q∗ 0 0    WeshallusethemtosimplifyranksandinertiasofblockmatricesinvolvingMoore–Penroseinverses of matrices. Lemma 1.10 ([2]) Let A, B ∈Cm×n be given. Then, the following hold. (a) Eq. (1.1) has a solution X ∈Cn if and only if R(B)⊆R(A) and AB∗ =BA∗. In this case, H the general Hermitian solution of (1.1) can be written as X =A†B+(A†B)∗−A†BA†A+F UF , (1.37) A A where U ∈Cn is arbitrary. H (b) The matrix equation in (1.1) has a solution 0 4 X ∈ Cn if and only if R(B) ⊆ R(A), H AB∗ <0 and R(AB∗)=R(BA∗)=R(B). In this case, the general solution 04X ∈Cn of H (1.1) can be written as X =B∗(AB∗)†B+F UF , (1.38) A A where 04U ∈Cn is arbitrary. H Lemma 1.11 Eq. (1.2) has a solution X ∈ Cn if and only if R(B) ⊆ R(A), or equivalently, H AA†B =B. In this case, the general Hermitian solution of AXA∗ =B can be written as X =A†B(A†)∗+F U +U∗F , (1.39) A A where U ∈Cn×n is arbitrary. 4 Lemma 1.12 ([11]) Let Aj ∈Cmj×n and Bj ∈CmHj be given, j =1, 2, and assume that A X A∗ =B and A X A∗ =B (1.40) 1 1 1 1 2 2 2 2 are solvable for X , X ∈Cn. Also define 1 2 H B 0 A 1 1 S = X ∈Cn | A X A∗ =B , j =1, 2, M = 0 −B A . (1.41) j j H j j j j  2 2 A∗ A∗ 0 (cid:8) (cid:9) 1 2   Then, max r(X −X )=min{n, r(M)+2n−2r(A )−2r(A )}, (1.42) 1 2 1 2 X1∈S1,X2∈S2 min r(X −X )=r(M)−2r[A∗, A∗], (1.43) 1 2 1 2 X1∈S1,X2∈S2 max i (X −X )=i (M)+n−r(A )−r(A ), (1.44) ± 1 2 ± 1 2 X1∈S1,X2∈S2 min i (X −X )=i (M)−r[A∗, A∗]. (1.45) ± 1 2 ± 1 2 X1∈S1,X2∈S2 Consequently, the following hold. (a) There exist X ∈ S and X ∈ S such that X −X is nonsingular if and only if r(M) > 1 1 2 2 1 2 2r(A )+2r(A )−n. 1 2 (b) X −X is nonsingular for all X ∈S and X ∈S if and only if r(M)=2r[A∗, A∗]+n. 1 2 1 1 2 2 1 2 (c) There exist X ∈ S and X ∈ S such that X = X if and only if R(B ) ⊆ R(A ) and 1 1 2 2 1 2 j j r(M)=2r[A∗, A∗], j =1, 2. 1 2 (d) The rank of X − X is invariant for all X ∈ S and X ∈ S if and only if r(M) = 1 2 1 1 2 2 2r[A∗, A∗]−n or r(A )=r(A )=n. 1 2 1 2 (e) There exist X ∈ S and X ∈ S such that X ≻ X (X ≺ X ) if and only if i (M) = 1 1 2 2 1 2 1 2 + r(A )+r(A ) (i (M)=r(A )+r(A )). 1 2 − 1 2 (f) X ≻ X (X ≺ X ) for all X ∈ S and X ∈ S if and only if i (M) = r[A∗, A∗]+n 1 2 1 2 1 1 2 2 + 1 2 (i (M)=r[A∗, A∗]+n). − 1 2 (g) There exist X ∈ S and X ∈ S such that X < X (X 4 X ) if and only if i (M) = 1 1 2 2 1 2 1 2 − r[A∗, A∗] (i (M)=r[A∗, A∗]). 1 2 + 1 2 (h) X

Equalities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations $AX = B$ and $AXA^* = B$

Similar Equalities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations $AX = B$ and $AXA^* = B$