Singular Values and Singular Value Inequalities 17-11 16.Let A Cmx".The following are equivalent: (a)o(AoB)≤om1(B)for all B∈Cx (⑥)∑1，(AoB)≤∑1，(B)for all B∈Cmx"and allk=l,…,q. (c)There are positive semidefinite PCxandQCsuch that is positive semidefinite,and has diagonal entries at most 1. 17.(Singular values and matrix entries)Take ACmx".Then (,....)((A)....(A).0.....). ws 子 lalP,0≤p≤2 2∑as2a.2sp<e. i=l i=l i=l iand column j of Aare 之an≥0anda1≥…≥an≥0.Thcn 3A∈R"xmst.:(A)=and ci(A)=ci÷(a,.,c)≤(o,,o) Π(A≤Πc(w.k=1,2…n The casek=1 is Hadamard's Inequality:I det(A)c(A). 20.[Tho77]Take F =Cor R and di,...d F such that dl only if 21.(Nonnegative matrices)Take A=[a]Cx". (a)If B=[lall,then a(A)1(B). (b)If A and B are real and0≤a≤bi,ji,then a(A)≤(B).The condition0≤aji essential.(Example) (c)The,j does not imply (AoB)<(A).(Example 4) 22.(Bound on g)Let A eC"".Then IlAll=(A)<AllAll 23.[Zha99](Cartesian decomposition)Let C=A+iB C"x",where A and B are Hermitian.Let A,B,C have singular valuesj=1...n.Then (h,a)≤wV2(la1+ig1l.…,lan+Bl)≤w2(h,,%). Singular Values and Singular Value Inequalities 17-11 16. Let A ∈ Cm×n. The following are equivalent: (a) σ1(A ◦ B) ≤ σ1(B) for all B ∈ Cm×n. (b) k i=1 σi(A ◦ B) ≤ k i=1 σi(B) for all B ∈ Cm×n and all k = 1, ... , q. (c) There are positive semidefinite P ∈ Cn×n and Q ∈ Cm×m such that
P A A∗ Q is positive semidefinite, and has diagonal entries at most 1. 17. (Singular values and matrix entries) Take A ∈ Cm×n. Then  |a11| 2 , |a12| 2 , ... , |amn| 2   σ2 1 (A), ... , σ2 q (A), 0, ... , 0 , q i=1 σ p i (A) ≤ m i=1 n j=1 |ai j| p , 0 ≤ p ≤ 2, m i=1 n j=1 |ai j| p ≤ q i=1 σ p i (A), 2 ≤ p < ∞. If σ1(A) = |ai j|, then all the other entries in row i and column j of A are 0. 18. Take σ1 ≥···≥ σn ≥ 0 and α1 ≥···≥ αn ≥ 0. Then ∃A ∈ Rn×n s.t. σi(A) = σi and ci(A) = αi ⇔  α2 1 , ... , α2 n    σ2 1 , ... , σ2 n  . This statement is still true if we replace Rn×n by Cn×n and/or ci( · ) by ri( · ). 19. Take A ∈ Cn×n. Then n i=k σi(A) ≤ n i=k ci(A), k = 1, 2, ... , n. The case k = 1 is Hadamard’s Inequality: | det(A)| ≤ n i=1 ci(A). 20. [Tho77] Take F = C or R and d1, ... , dn ∈ F such that |d1|≥···≥|dn|, and σ1 ≥···≥ σn ≥ 0. There is a matrix A ∈ F n×n with diagonal entries d1, ... , dn and singular values σ1, ... , σn if and only if (|d1|, ... , |dn|) w (σ1(A), ... , σn(A)) and n−1 j=1 |dj|−|dn| ≤ n−1 j=1 σj(A) − σn(A). 21. (Nonnegative matrices) Take A = [ai j] ∈ Cm×n. (a) If B = [|ai j|], then σ1(A) ≤ σ1(B). (b) If A and B are real and 0 ≤ ai j ≤ bi j ∀ i, j, then σ1(A) ≤ σ1(B). The condition 0 ≤ ai j is essential. (Example 4) (c) The condition 0 ≤ bi j ≤ 1 ∀ i, j does not imply σ1(A ◦ B) ≤ σ1(A). (Example 4) 22. (Bound on σ1) Let A ∈ Cm×n. Then A2 = σ1(A) ≤ √A1A∞. 23. [Zha99] (Cartesian decomposition) Let C = A + i B ∈ Cn×n, where A and B are Hermitian. Let A, B,C have singular values αj , βj , γi , j = 1, ... , n. Then (γ1, ... , γn) w √ 2(|α1 + iβ1|, ... , |αn + iβn|) w 2(γ1, ... , γn).