v2007.09.13 - Convex Optimization

480 APPENDIX A. LINEAR ALGEBRAA.1.1IdentitiesThis δ notation is efficient and unambiguous as illustrated in the followingexamples where A ◦ B denotes Hadamard product [149] [109,1.1.4] ofmatrices of like size, ⊗ the Kronecker product (D.1.2.1), y a vector, X amatrix, e i the i th member of the standard basis for R n , S N h the symmetrichollow subspace, σ a vector of (nonincreasingly) ordered singular values, andλ(A) denotes a vector of nonincreasingly ordered eigenvalues of matrix A :1. δ(A) = δ(A T )2. tr(A) = tr(A T ) = δ(A) T 13. 〈I, A〉 = trA4. δ(cA) = cδ(A), c∈R5. tr(c √ A T A) = c tr √ A T A = c1 T σ(A), c∈R6. tr(cA) = c tr(A) = c1 T λ(A), c∈R7. δ(A + B) = δ(A) + δ(B)8. tr(A + B) = tr(A) + tr(B)9. δ(AB) = (A ◦ B T )1 = (B T ◦ A)110. δ(AB) T = 1 T (A T ◦ B) = 1 T (B ◦ A T )⎡ ⎤u 1 v 111. δ(uv T ⎢ ⎥) = ⎣ . ⎦ = u ◦ v , u,v ∈ R Nu N v N12. tr(A T B) = tr(AB T ) = tr(BA T ) = tr(B T A)= 1 T (A ◦ B)1 = 1 T δ(AB T ) = δ(A T B) T 1 = δ(BA T ) T 1 = δ(B T A) T 113. D = [d ij ] ∈ S N h , H = [h ij ] ∈ S N h , V = I − 1 N 11T ∈ S N (conferB.4.2 no.20)N tr(−V (D ◦ H)V ) = tr(D T H) = 1 T (D ◦ H)1 = tr(11 T (D ◦ H)) = ∑ d ij h iji,j14. tr(ΛA) = δ(Λ) T δ(A), δ 2 (Λ) ∆ = Λ ∈ S N