v2010.10.26 - Convex Optimization

2.2. VECTORIZED-MATRIX INNER PRODUCT 51vectorization. For example, the vectorization of Y = [y 1 y 2 · · · y k ] ∈ R p×k[166] [327] is⎡ ⎤y 1yvec Y ⎢ 2⎥⎣ . ⎦ ∈ Rpk (37)y kThen the vectorized-matrix inner-product is trace of matrix inner-product;for Z ∈ R p×k , [61,2.6.1] [199,0.3.1] [382,8] [365,2.2]where (A.1.1)〈Y , Z〉 tr(Y T Z) = vec(Y ) T vec Z (38)tr(Y T Z) = tr(ZY T ) = tr(YZ T ) = tr(Z T Y ) = 1 T (Y ◦ Z)1 (39)and where ◦ denotes the Hadamard product 2.11 of matrices [159,1.1.4]. Theadjoint A T operation on a matrix can therefore be defined in like manner:〈Y , A T Z〉 〈AY , Z〉 (40)Take any element C 1 from a matrix-valued set in R p×k , for example, andconsider any particular dimensionally compatible real vectors v and w .Then vector inner-product of C 1 with vw T is〈vw T , C 1 〉 = 〈v , C 1 w〉 = v T C 1 w = tr(wv T C 1 ) = 1 T( (vw T )◦ C 1)1 (41)Further, linear bijective vectorization is distributive with respect toHadamard product; id est,vec(Y ◦ Z) = vec(Y ) ◦ vec(Z) (42)2.2.0.0.1 Example. Application of inverse image theorem.Suppose set C ⊆ R p×k were 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 (43)2.11 Hadamard product is a simple entrywise product of corresponding entries from twomatrices of like size; id est, not necessarily square. A commutative operation, theHadamard product can be extracted from within a Kronecker product. [202, p.475]