Notice that if a is i× k and b is m×n, then A &B is(im)×(km) Useful results 1. Let A, B, C, and D be any matrices, then 1 (A⑧B)(C⑧D)=(AC)②(BD), (A+B)⑧C=(A⑧C)+(B⑧C), 1 (A⑧B)=A⑧B′ 2. Let a be m× m and b be k× k matrices,then 2a &B A-1⑧B Proof: (A-1B-1)(A⑧B)=(AA⑧B-1B)=Im⑧Ik=I (A⑧B)=tr(A)tr(B) Proof: tr(AgB)=2atr(B)=tr(A)tr(B) 4.5 The Vec Operator The operator that transforms a matrix to a vector is known as the vec operator If the m x n matrix A has ai as its ith column, then vec(A)is the mn x 1 vectors veNotice that if A is i × k and B is m × n, then A ⊗ B is (im) × (kn). Useful results: 1. Let A, B, C, and D be any matrices, then 1a. (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD), 1b. (A + B) ⊗ C = (A ⊗ C) + (B ⊗ C), 1c. (A ⊗ B) ′ = A′ ⊗ B ′ . 2. Let A be m × m and B be k × k matrices, then 2a. (A ⊗ B) −1 = A−1 ⊗ B −1 . Proof: (A−1 ⊗ B −1 )(A ⊗ B) = (A−1A ⊗ B −1B) = Im ⊗ Ik = Imk. 2b. tr(A ⊗ B) = tr(A)tr(B). Proof: tr(A ⊗ B) = Xm i=1 aiitr(B) = tr(A)tr(B). 4.5 The Vec Operator The operator that transforms a matrix to a vector is known as the vec operator. If the m×n matrix A has ai as its ith column, then vec(A) is the mn×1 vectors given by vec(A) =         a1 a2 . . . an         . 18