Singular Values and Singular Value Inequalities 17-13 In particular,for the spectral norm and the Frobenius norm,we have x+(A)min(llA Bll2 rank(B)k), (三2w 2 =min(llA -BlF rank(B)k). IA-Ulvr≤IA-Wlor≤IA+UIu 3.[GV96,$12.4.1][H85,Ex.7.4.8](Orthogonal Procrustes problem)Let A,BCmx Let BA have a polar decomposition B*A =UP.Then IlA-BUllE min(IA-BWllF WEC"x",w*w=1). This app AH =(A+A)2,B=(An+)/2) Then B is positive semidefinite and is the min(lA-XIF:X∈Cxm,X∈PSDl appro gular value d sition Uand Vbe any unitary matrices.Then IEA-BlI≤A-UBV*I 17.6 Characterization of the Eigenvalues of Sums of Hermitian Matrices and Singular Values of Sums and Products of General Matrices There are necessary and sufficient conditions for three sets of numbers to be the eigenvalues of Hermitian 众C二i6G ([Kly98])and Knutson and Tao([KT99]).The results presented here are from a survey by Fulton [Ful0o]. Bhatia has written an expository paper on the subject([Bha)). Definitions: Theinequalitiesare interms ofthe setsT"oftriples(,,K)ofsubsetsof(,...,nofthe same cardinality r,defined by the following inductive procedure.Set =1.是+g1=爱++w Whenr=1,set T"=U".In general, T"(1,J,K)U"Ifor all p<r and all (E,G,用i血∑+∑k≤∑+pp+/p In this section,the vectors,B,y will have real entries ordered in nonincreasing order. Singular Values and Singular Value Inequalities 17-13 In particular, for the spectral norm and the Frobenius norm, we have σk+1(A) = min{A − B2 : rank(B) ≤ k},  q i=k+1 σ2 k+1(A) 1/2 = min{A − BF : rank(B) ≤ k}. 2. [Bha97, p. 276] (Best unitary approximation) Take A, W ∈ Cn×n with W unitary. Let A = UP be a polar decomposition of A. Then A − UU I ≤ A − WU I ≤ A + UU I . 3. [GV96, §12.4.1] [HJ85, Ex. 7.4.8] (Orthogonal Procrustes problem) Let A, B ∈ Cm×n. Let B∗A have a polar decomposition B∗A = U P . Then A − BUF = min{A − B WF : W ∈ Cn×n, W∗W = I}. This result is not true if ·F is replaced by ·U I ([Mat93, §4]). 4. [Hig89] (Best PSD approximation) Take A ∈ Cn×n. Set AH = (A + A∗)/2, B = (AH + |AH |)/2). Then B is positive semidefinite and is the unique solution to min{A − XF : X ∈ Cn×n, X ∈ PSD}. There is also a formula for the best PSD approximation in the spectral norm. 5. Let A, B ∈ Cm×n have singular value decompositions A = UAAV∗ A and B = UBBV∗ B . Let U ∈ Cm×m and V ∈ Cn×n be any unitary matrices. Then A − B U I ≤ A − UBV∗U I . 17.6 Characterization of the Eigenvalues of Sums of Hermitian Matrices and Singular Values of Sums and Products of General Matrices There are necessary and sufficient conditions for three sets of numbers to be the eigenvalues of Hermitian A, B,C = A + B ∈ Cn×n, or the singular values of A, B,C = A + B ∈ Cm×n, or the singular values of nonsingular A, B, C = AB ∈ Cn×n. The key results in this section were first proved by Klyachko ([Kly98]) and Knutson and Tao ([KT99]). The results presented here are from a survey by Fulton [Ful00]. Bhatia has written an expository paper on the subject ([Bha01]). Definitions: The inequalities are in terms of the sets Tn r of triples (I, J , K) of subsets of{1, ... , n} of the same cardinality r, defined by the following inductive procedure. Set Un r =  (I, J , K)      i∈I i + j∈J j = k∈K k + r(r + 1)/2  . When r = 1, set Tn 1 = Un 1 . In general, Tn r =  (I, J , K) ∈ Un r | for all p < r and all (F, G, H) in Tr p, f∈F if + g∈G jg ≤ h∈H kh + p(p + 1)/2  . In this section, the vectors α, β, γ will have real entries ordered in nonincreasing order.