v2006.03.09 - Convex Optimization

A.3. PROPER STATEMENTS 435For A∈ R m×n having no nullspace, and for any B ∈ R n×krank(AB) = rank(B) (1072)Proof. For any compatible matrix C , N(CAB)⊇ N(AB)⊇ N(B)is obvious. By assumption ∃A † A † A = I . Let C = A † , thenN(AB)= N(B) and the stated result follows by conservation ofdimension (1173).For A∈ S n and any nonsingular matrix Yinertia(A) = inertia(YAY T ) (1073)a.k.a, Sylvester’s law of inertia. (1114) [61,2.4.3]For A,B∈R n×n square,Yet for A∈ R m×n and B ∈ R n×m [44, p.72]det(AB) = det(BA) (1074)det(I + AB) = det(I + BA) (1075)For A,B ∈ S n , product AB is symmetric if and only if AB iscommutative;(AB) T = AB ⇔ AB = BA (1076)Proof. (⇒) Suppose AB=(AB) T . (AB) T =B T A T =BA .AB=(AB) T ⇒ AB=BA .(⇐) Suppose AB=BA . BA=B T A T =(AB) T . AB=BA ⇒AB=(AB) T .Commutativity alone is insufficient for symmetry of the product.[212, p.26] Diagonalizable matrices A,B ∈ R n×n commute if and onlyif they are simultaneously diagonalizable. [125,1.3.12]