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.