17-6 Handbook of Linear Algebra The trace norm of ACmx"is 一discusednth邮ction.suc由as the spectral norm-l0A=oMA=mag） and the Frobenius Warning:There is potential for considerable confusion.For example,llAll2 =llAllk=Alls,while l.llso unless m=1),and generally llAll2,llAlls.2 and llAllk.2 are all different,as are llAll, Alls.1 and llAllk.I.Nevertheless,many authors use ll lk for lllx&and lp for lllls.p. Facts: The following standard facts can be found in many texts,eg(H91,5]and [Bha97,Chap.IV]. 1.It is unitarily invariant ifand only if there isa permutation invariant ,ogA)for all A∈Cm nd letg bet ponding permutation invariant the dual norms(se 3H9 Chapter 37) 801As·0g1) ma,cep+p rm are d while the =,A.1... 4.For amy 5.Ifll.lis F≤IA2≤IA on C,then N(A) A+a u.i.norm on Cx.A norm that arises in this way is called a 6.Let A,BCx be given.The following areequivalent (a)AluB for all unitarily invariant normsv (b)IlAllk s llBlK&fork =1,2.....q. ()(a(A,.,g(A)≤w(a(B,…,og(B)l.(≤is defined in Preliminaries) The equivalence of the first two conditions is Fan's Dominance Theorem. 7.The Ky-Fan-k norms can be represented in terms of an extremal problem involving the spectral norm and the trace norm.Take A C"x".Then llAllK&minllxlr+:+Y=A)k=1,....q. 8.[HJ91,Theorem 3.3.14]Take A,B Cmx".Then This isan important result in developing the theory of unitarily invariant norms
17-6 Handbook of Linear Algebra The trace norm of A ∈ Cm×n is Atr = q i=1 σi(A) = AK,q = AS,1 = tr |A|pd . Other norms discussed in this section, such as the spectral norm ·2 (A2 = σ1(A) = maxx=0 Ax2 x2 ) and the Frobenius norm ·F (AF = ( q i=1 σ2 i (A))1/2 = ( m i=1 n j=1 |ai j| 2)1/2), are defined in Section 7.1. and discussed extensively in Chapter 37. Warning: There is potential for considerable confusion. For example, A2 = AK,1 = AS,∞, while ·∞ =·S,∞ ( unless m = 1), and generally A2, AS,2 and AK,2 are all different, as are A1, AS,1 and AK,1. Nevertheless, many authors use ·k for ·K,k and ·p for ·S,p . Facts: The following standard facts can be found in many texts, e.g., [HJ91, §3.5] and [Bha97, Chap. IV]. 1. Let · be a norm on Cm×n. It is unitarily invariant if and only if there is a permutation invariant absolute norm g on Rq such that A = g (σ1(A), ... , σq (A)) for all A ∈ Cm×n. 2. Let · be a unitarily invariant norm onCm×n, and let g be the corresponding permutation invariant absolute norm g . Then the dual norms (see Chapter 37) satisfy AD = g D(σ1(A), ... , σq (A)). 3. [HJ91, Prob. 3.5.18] The spectral norm and trace norm are duals, while the Frobenius norm is self dual. The dual of ·S,p is ·S,p˜ , where 1/p + 1/p˜ = 1 and AD K,k = max A2, Atr k , k = 1, ... , q. 4. For any A ∈ Cm×n, q−1/2AF ≤ A2 ≤ AF . 5. If · is a u.i. norm on Cm×n, then N(A) = A∗A1/2 is a u.i. norm on Cn×n. A norm that arises in this way is called a Q-norm. 6. Let A, B ∈ Cm×n be given. The following are equivalent (a) AU I ≤ BU I for all unitarily invariant norms ·U I . (b) AK,k ≤ BK,k for k = 1, 2, ... , q. (c) (σ1(A), ... , σq (A)) w (σ1(B), ... , σq (B)). ( w is defined in Preliminaries) The equivalence of the first two conditions is Fan’s Dominance Theorem. 7. The Ky–Fan-k norms can be represented in terms of an extremal problem involving the spectral norm and the trace norm. Take A ∈ Cm×n. Then AK,k = min{Xtr + kY2 : X + Y = A} k = 1, ... , q. 8. [HJ91, Theorem 3.3.14] Take A, B ∈ Cm×n. Then |trAB∗| ≤ q i=1 σi(A)σi(B). This is an important result in developing the theory of unitarily invariant norms.
Singular Values and Singular Value Inequalities 17-7 Examples: 1.The matrix A in Example 1 of Section 17.1 has singular values20,128,and 4.So IIAll2 =20,IlAllF =24,IlAll =44; 1AlK.1=20,1Ax2=32,IAK.3=40,1AK4=44 1Als.1=44,1As2=√624，IAls3=√/10304=21.7605,1A5.x=20. 17.4 Inequalities Definitions: Pinching is defined recursively.If A21 A22 of B is Facts: T ca be foundntdard reeenc,or eample 9 Chap.unkes anothe 1.(Submatrices)Take AC and let B denote A with one of its rows or columns deleted.Then 之A公om dod.Then o42(A)≤o(B)≤0(A),i=1,...,9-2 Thei+2 ca B be an (m-k)x(n-1)submatrix of A.Then O+k+H(A)≤a(B)≤o(A,i=1,,q-(k+I). 4.Take AC and let B be A with some of its rows and/or columns set to zero.Then(B)s A) 1 5.Let B bea qualitiesΠ1(B)≤o(A)and (B)< not n arily true fork>1.(Example 1) 6.(Singular values ofA+B)Let A.BC (a)sv(A+B)sv(A)+sv(B),or equivalently ∑o(A+B)≤∑o,(A)+∑0(B,i=1,,9 i=1 (b)Ifi+j-1s q andi,jEN,then (A+B)s(A)+aj(B)
Singular Values and Singular Value Inequalities 17-7 Examples: 1. The matrix A in Example 1 of Section 17.1 has singular values 20, 12, 8, and 4. So A2 = 20, AF = √624, Atr = 44; AK,1 = 20, AK,2 = 32, AK,3 = 40, AK,4 = 44; AS,1 = 44, AS,2 = √624, AS,3 = √3 10304 = 21.7605, AS,∞ = 20. 17.4 Inequalities Throughout this section, q = min{m, n} and if A ∈ Cm×n has real eigenvalues, then they are ordered λ1(A) ≥···≥ λn(A). Definitions: Pinching is defined recursively. If A = A11 A12 A21 A22 ∈ Cm×n, B = A11 0 0 A22 ∈ Cm×n, then B is a pinching of A. (Note that we do not require the Aii to be square.) Furthermore, any pinching of B is a pinching of A. For positive α, β, define the measure of relative separation χ(α, β) = |√α/β − √β/α|. Facts: The following facts can be found in standard references, for example [HJ91, Chap. 3], unless another reference is given. 1. (Submatrices) Take A ∈ Cm×n and let B denote A with one of its rows or columns deleted. Then σi+1(A) ≤ σi(B) ≤ σi(A), i = 1, ... , q − 1. 2. Take A ∈ Cm×n and let B be A with a row and a column deleted. Then σi+2(A) ≤ σi(B) ≤ σi(A), i = 1, ... , q − 2. The i + 2 cannot be replaced by i + 1. (Example 2) 3. Take A ∈ Cm×n and let B be an (m − k) × (n − l) submatrix of A. Then σi+k+l(A) ≤ σi(B) ≤ σi(A), i = 1, ... , q − (k + l). 4. Take A ∈ Cm×n and let B be A with some of its rows and/or columns set to zero. Then σi(B) ≤ σi(A), i = 1, ... , q. 5. Let B be a pinching of A. Then sv(B) w sv(A). The inequalities k i=1 σi(B) ≤ k i=1 σi(A) and σk (B) ≤ σk (A) are not necessarily true for k > 1. (Example 1) 6. (Singular values of A + B) Let A, B ∈ Cm×n. (a) sv(A + B) w sv(A) + sv(B), or equivalently k i=1 σi(A + B) ≤ k i=1 σi(A) + k i=1 σi(B), i = 1, ... , q. (b) If i + j − 1 ≤ q and i, j ∈ N, then σi+ j−1(A + B) ≤ σi(A) + σj(B).