v2010.10.26 - Convex Optimization

594 APPENDIX A. LINEAR ALGEBRA35. For any permutation matrix Ξ and dimensionally compatible vector yor matrix Aδ(Ξy) = Ξδ(y) Ξ T (1419)δ(ΞAΞ T ) = Ξδ(A) (1420)So given any permutation matrix Ξ and any dimensionally compatiblematrix B , for example,δ 2 (B) = Ξδ 2 (Ξ T B Ξ)Ξ T (1421)36. A ⊗ 1 = 1 ⊗ A = A37. A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C38. (A ⊗ B)(C ⊗ D) = AC ⊗ BD39. For A a vector, (A ⊗ B) = (A ⊗ I)B40. For B a row vector, (A ⊗ B) = A(I ⊗ B)41. (A ⊗ B) T = A T ⊗ B T42. (A ⊗ B) −1 = A −1 ⊗ B −143. tr(A ⊗ B) = trA trB44. For A∈ R m×m , B ∈ R n×n , det(A ⊗ B) = det n (A) det m (B)45. There exist permutation matrices Ξ 1 and Ξ 2 such that [166, p.28]A ⊗ B = Ξ 1 (B ⊗ A)Ξ 2 (1422)46. For eigenvalues λ(A)∈ C n and eigenvectors v(A)∈ C n×n such thatA = vδ(λ)v −1 ∈ R n×nλ(A ⊗ B) = λ(A) ⊗ λ(B) , v(A ⊗ B) = v(A) ⊗ v(B) (1423)47. Given analytic function [82] f : C n×n → C n×n , then f(I ⊗A)= I ⊗f(A)and f(A ⊗ I) = f(A) ⊗ I [166, p.28]