Singular Values and Singular Value Inequalities 17-9 (b)If AB is Hermitian,then sv(AB)sv(H(BA))and llABllur H(BA)Ilui,where H(X)=(X+X)/2. 9.(Term-wise singular value inequalities)[Zha2,p.28]Take A,BCm.Then 20;(AB")s i(A*A+B*B),i=1.....q and,more generally,if p,and 1/p+/=1,then 0:(AB*)<g: 、p The2a(A'B)≤a(A+B产B）and(A+B)≤(Al+Bl)are not tru in general (Example 3),but we do have IA*BI呢，≤A*Allv:lB*BuI 2he97.op.m5 Tke Ae C9ThmA十4≤2aw.i=2 11.[LM02](Block triangular matrices)Let A S T ∈Cx"(R∈CpxP)have singular values a之…之aa.Lctk=min(p,n-pl.Thcn (a)If omin(R)2Omax(T),then o(R)≤，i=l,,p ≤i-p(T,i=p+1…,n b)(a1(S).,0m(S》≤(@1-m…,k-a-k+i (c)If A is invertible,then (a(T-1SR-l,,m(T-lSR-l)≤w(a-a,…,a+1-a）, (-4-） 卫Log(d)-[人Ag Ai2 Az2 Cxbe positive definite with eigenvalues≥…≥x Assume An∈Cpxp.Setk=minip,n-pl.Then IiA)≤1(Ana(Aahj=lk (a(APA2,m(APA)≤w(V万-VV-Va-+ (a1(4'A2,ax(4A2)》≤w)(xa,n,X2b入m-k+) Ifk =n/2,then Anllrs Anllur IAzllur. l3.(Singular values and eigenvalues))LetA∈Cx.Assume1(Al≥…≥ln(Al.Then (a)Π，l(Al≤Πtm(A,k=l,,m,with equality fork=m. Singular Values and Singular Value Inequalities 17-9 (b) If AB is Hermitian, then sv(AB) w sv(H(B A)) and ABU I ≤ H(B A)U I , where H(X) = (X + X∗)/2. 9. (Term-wise singular value inequalities) [Zha02, p. 28] Take A, B ∈ Cm×n. Then 2σi(AB∗) ≤ σi(A∗A + B∗B), i = 1, ... , q and, more generally, if p, p˜ > 0 and 1/p + 1/p˜ = 1, then σi(AB∗) ≤ σi  (A∗A)p/2 p + (B∗B)p˜/2 p˜  = σi |A| p pd p + |B| p˜ pd p˜  . The inequalities 2σ1(A∗B) ≤ σ1(A∗A + B∗B) and σ1(A + B) ≤ σ1(|A|pd + |B|pd ) are not true in general (Example 3), but we do have A∗B2 U I ≤ A∗AU I B∗BU I . 10. [Bha97, Prop. III.5.1] Take A ∈ Cn×n. Then λi(A + A∗) ≤ 2σi(A), i = 1, 2, ... , n. 11. [LM02] (Block triangular matrices) Let A = ⎡ ⎣ R 0 S T ⎤ ⎦ ∈ Cn×n (R ∈ Cp×p ) have singular values α1 ≥···≥ αn. Let k = min{p, n − p}. Then (a) If σmin(R) ≥ σmax(T), then σi(R) ≤ αi , i = 1, ... , p αi ≤ σi−p (T), i = p + 1, ... , n. (b) (σ1(S), ... , σk (S)) w (α1 − αn, ··· , αk − αn−k+1). (c) If A is invertible, then (σ1(T−1 S R−1 , ... , σk (T−1 S R−1 ) w  α−1 n − α−1 1 , ··· , α−1 n−k+1 − α−1 k  , (σ1(T−1 S), ... , σk (T−1 S)) w 1 2  α1 αn − αn α1 , ··· , αk αn−k+1 − αn−k+1 αk  . 12. [LM02] (Block positive semidefinite matrices) Let A = ⎡ ⎣ A11 A12 A∗ 12 A22 ⎤ ⎦ ∈ Cn×n be positive definite with eigenvalues λ1 ≥···≥ λn. Assume A11 ∈ Cp×p . Set k = min{p, n − p}. Then  j i=1 σ2 i (A12) ≤  j i=1 σi(A11)σi(A22), j = 1, ... , k,  σ1  A−1/2 11 A12 , ... , σk  A−1/2 11 A12 w λ1 − λn, ... , λk − λn−k+1  ,  σ1  A−1 11 A12 , ... , σk  A−1 11 A12 w 1 2 (χ(λ1, λn), ... , χ(λk , λn−k+1)). If k = n/2, then A122 U I ≤ A11U I A22U I . 13. (Singular values and eigenvalues) Let A ∈ Cn×n. Assume |λ1(A)|≥···≥|λn(A)|. Then (a) k i=1 |λi(A)| ≤ k i=k σi(A), k = 1, ... , n, with equality for k = n