v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
592 APPENDIX A. LINEAR ALGEBRAA.1.1IdentitiesThis δ notation is efficient and unambiguous as illustrated in the followingexamples where: A ◦ B denotes Hadamard product [202] [159,1.1.4] ofmatrices of like size, ⊗ Kronecker product [166] (D.1.2.1), y a vector, Xa matrix, e i the i th member of the standard basis for R n , S N h the symmetrichollow subspace, σ(A) a vector of (nonincreasingly) ordered singular values,and λ(A) a vector of nonincreasingly ordered eigenvalues of matrix A :1. δ(A) = δ(A T )2. tr(A) = tr(A T ) = δ(A) T 1 = 〈I, A〉3. δ(cA) = cδ(A) c∈R4. tr(cA) = c tr(A) = c1 T λ(A) c∈R5. vec(cA) = c vec(A) c∈R6. A ◦ cB = cA ◦ B c∈R7. A ⊗ cB = cA ⊗ B c∈R8. δ(A + B) = δ(A) + δ(B)9. tr(A + B) = tr(A) + tr(B)10. vec(A + B) = vec(A) + vec(B)11. (A + B) ◦ C = A ◦ C + B ◦ CA ◦ (B + C) = A ◦ B + A ◦ C12. (A + B) ⊗ C = A ⊗ C + B ⊗ CA ⊗ (B + C) = A ⊗ B + A ⊗ C13. sgn(c)λ(|c|A) = cλ(A) c∈R14. sgn(c)σ(|c|A) = cσ(A) c∈R15. tr(c √ A T A) = c tr √ A T A = c1 T σ(A) c∈R16. π(δ(A)) = λ(I ◦A) where π is the presorting function.17. δ(AB) = (A ◦ B T )1 = (B T ◦ A)118. δ(AB) T = 1 T (A T ◦ B) = 1 T (B ◦ A T )
A.1. MAIN-DIAGONAL δ OPERATOR, λ , TRACE, VEC 593⎡ ⎤u 1 v 119. δ(uv T ⎢ ⎥) = ⎣ . ⎦ = u ◦ v ,u N v Nu,v ∈ R N20. 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 121. 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,j22. tr(ΛA) = δ(Λ) T δ(A), δ 2 (Λ) Λ ∈ S N23. y T Bδ(A) = tr ( Bδ(A)y T) = tr ( δ(B T y)A ) = tr ( Aδ(B T y) )= δ(A) T B T y = tr ( yδ(A) T B T) = tr ( A T δ(B T y) ) = tr ( δ(B T y)A T)24. δ 2 (A T A) = ∑ ie i e T iA T Ae i e T i25. δ ( δ(A)1 T) = δ ( 1δ(A) T) = δ(A)26. δ(A1)1 = δ(A11 T ) = A1 , δ(y)1 = δ(y1 T ) = y27. δ(I1) = δ(1) = I28. δ(e i e T j 1) = δ(e i ) = e i e T i29. For ζ =[ζ i ]∈ R k and x=[x i ]∈ R k ,30.∑ζ i /x i = ζ T δ(x) −1 1vec(A ◦ B) = vec(A) ◦ vec(B) = δ(vecA) vec(B)= vec(B) ◦ vec(A) = δ(vecB) vec(A)31. vec(AXB) = (B T ⊗ A) vec X32. vec(BXA) = (A T ⊗ B) vec Xi(42)(1787)(notH )33.34.tr(AXBX T ) = vec(X) T vec(AXB) = vec(X) T (B T ⊗ A) vec X [166]= δ ( vec(X) vec(X) T (B T ⊗ A) ) T1tr(AX T BX) = vec(X) T vec(BXA) = vec(X) T (A T ⊗ B) vec X= δ ( vec(X) vec(X) T (A T ⊗ B) ) T1
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592 APPENDIX A. LINEAR ALGEBRAA.1.1IdentitiesThis δ notation is efficient and unambiguous as illustrated in the followingexamples where: A ◦ B denotes Hadamard product [202] [159,1.1.4] ofmatrices of like size, ⊗ Kronecker product [166] (D.1.2.1), y a vector, Xa matrix, e i the i th member of the standard basis for R n , S N h the symmetrichollow subspace, σ(A) a vector of (nonincreasingly) ordered singular values,and λ(A) a vector of nonincreasingly ordered eigenvalues of matrix A :1. δ(A) = δ(A T )2. tr(A) = tr(A T ) = δ(A) T 1 = 〈I, A〉3. δ(cA) = cδ(A) c∈R4. tr(cA) = c tr(A) = c1 T λ(A) c∈R5. vec(cA) = c vec(A) c∈R6. A ◦ cB = cA ◦ B c∈R7. A ⊗ cB = cA ⊗ B c∈R8. δ(A + B) = δ(A) + δ(B)9. tr(A + B) = tr(A) + tr(B)10. vec(A + B) = vec(A) + vec(B)11. (A + B) ◦ C = A ◦ C + B ◦ CA ◦ (B + C) = A ◦ B + A ◦ C12. (A + B) ⊗ C = A ⊗ C + B ⊗ CA ⊗ (B + C) = A ⊗ B + A ⊗ C13. sgn(c)λ(|c|A) = cλ(A) c∈R14. sgn(c)σ(|c|A) = cσ(A) c∈R15. tr(c √ A T A) = c tr √ A T A = c1 T σ(A) c∈R16. π(δ(A)) = λ(I ◦A) where π is the presorting function.17. δ(AB) = (A ◦ B T )1 = (B T ◦ A)118. δ(AB) T = 1 T (A T ◦ B) = 1 T (B ◦ A T )
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