v2006.03.09 - Convex Optimization

58 CHAPTER 2. CONVEX GEOMETRYand where ◦ denotes the Hadamard product 2.9 of matrices [125] [88,1.1.4].The adjoint operation A T on a matrix can therefore be defined in like manner:〈Y , A T Z〉 ∆ = 〈AY , Z〉 (32)For example, take any element C 1 from a matrix-valued set in R p×k ,and consider any particular dimensionally compatible real vectors v andw . Then vector inner-product of C 1 with vw T is〈vw T , C 1 〉 = v T C 1 w = tr(wv T C 1 ) = 1 T( (vw T )◦ C 1)1 (33)2.2.0.0.1 Example. Application of the image theorem.Suppose the set C ⊆ R p×k is convex. Then for any particular vectors v ∈R pand w ∈R k , the set of vector inner-productsY ∆ = v T Cw = 〈vw T , C〉 ⊆ R (34)is convex. This result is a consequence of the image theorem. Yet it is easyto show directly that convex combination of elements from Y remains anelement of Y . 2.10More generally, vw T in (34) may be replaced with any particular matrixZ ∈ R p×k while convexity of the set 〈Z , C〉⊆ R persists. Further, byreplacing v and w with any particular respective matrices U and W ofdimension compatible with all elements of convex set C , then set U T CWis convex by the image theorem because it is a linear mapping of C .2.9 The Hadamard product is a simple entrywise product of corresponding entries fromtwo matrices of like size; id est, not necessarily square.2.10 To verify that, take any two elements C 1 and C 2 from the convex matrix-valued setC , and then form the vector inner-products (34) that are two elements of Y by definition.Now make a convex combination of those inner products; videlicet, for 0≤µ≤1µ 〈vw T , C 1 〉 + (1 − µ) 〈vw T , C 2 〉 = 〈vw T , µ C 1 + (1 − µ)C 2 〉The two sides are equivalent by linearity of inner product. The right-hand side remainsa vector inner-product of vw T with an element µ C 1 + (1 − µ)C 2 from the convex set C ;hence it belongs to Y . Since that holds true for any two elements from Y , then it mustbe a convex set.