v2007.09.17 - Convex Optimization

50 CHAPTER 2. CONVEX GEOMETRYwhere all entries off the main diagonal have been scaled. Now for Z ∈ S M〈Y , Z〉 ∆ = tr(Y T Z) = vec(Y ) T vec Z = svec(Y ) T svec Z (48)Then because the metrics become equivalent, for X ∈ S M‖ svec X − svec Y ‖ 2 = ‖X − Y ‖ F (49)and because symmetric vectorization (47) is a linear bijective mapping, thensvec is an isometric isomorphism on the symmetric matrix subspace. In otherwords, S M is isometrically isomorphic with R M(M+1)/2 in the Euclidean senseunder transformation svec .The set of all symmetric matrices S M forms a proper subspace in R M×M ,so for it there exists a standard orthonormal basis in isometrically isomorphicR M(M+1)/2 {E ij ∈ S M } =⎧⎨⎩e i e T i ,1 √2(e i e T j + e j e T ii = j = 1...M), 1 ≤ i < j ≤ M⎫⎬⎭(50)where M(M +1)/2 standard basis matrices E ij are formed from the standardbasis vectors e i ∈ R M . Thus we have a basic orthogonal expansion for Y ∈ S MY =M∑ j∑〈E ij ,Y 〉 E ij (51)j=1 i=1whose coefficients〈E ij ,Y 〉 ={Yii , i = 1... MY ij√2 , 1 ≤ i < j ≤ M(52)correspond to entries of the symmetric vectorization (47).