v2009.01.01 - Convex Optimization

540 APPENDIX A. LINEAR ALGEBRA
29. For any permutation matrix Ξ and dimensionally compatible vector y
or matrix A
δ(Ξy) = Ξδ(y) Ξ T (1322)
δ(ΞAΞ T ) = Ξδ(A) (1323)
So given any permutation matrix Ξ and any dimensionally compatible
matrix B , for example,
δ 2 (B) = Ξδ 2 (Ξ T B Ξ)Ξ T (1324)
30. For κ a scalar, A ⊗ (κB) = κ(A ⊗ B)
31. (A + B) ⊗ C = A ⊗ C + B ⊗ C
A ⊗ (B + C) = A ⊗ B + A ⊗ C
32. A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C
33. (A ⊗ B)(C ⊗ D) = AC ⊗ BD
34. (A ⊗ B) T = A T ⊗ B T
35. (A ⊗ B) −1 = A −1 ⊗ B −1
36. tr(A ⊗ B) = trA trB
37. For A∈ R m×m , B ∈ R n×n , det(A ⊗ B) = det n (A) det m (B)
38. There exist permutation matrices Ξ 1 and Ξ 2 such that [140, p.28]
A ⊗ B = Ξ 1 (B ⊗ A)Ξ 2 (1325)
39. For eigenvalues λ(A)∈ C n and eigenvectors v(A)∈ C n×n such that
A = vδ(λ)v −1 ∈ R n×n
λ(A ⊗ B) = λ(A) ⊗ λ(B) , v(A ⊗ B) = v(A) ⊗ v(B) (1326)
40. Given analytic function [73] f : C n×n → C n×n , then f(I ⊗A)= I ⊗f(A)
and f(A ⊗ I) = f(A) ⊗ I [140, p.28]