v2007.09.13 - Convex Optimization

A.3. PROPER STATEMENTS 491Because R(A T A)= R(A T ) and R(AA T )= R(A) , for any A∈ R m×nrank(AA T ) = rank(A T A) = rankA (1269)For A∈ R m×n having no nullspace, and for any B ∈ R n×krank(AB) = rank(B) (1270)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 (1364).For A∈ S n and any nonsingular matrix Yinertia(A) = inertia(YAY T ) (1271)a.k.a, Sylvester’s law of inertia. (1312) [77,2.4.3]For A,B∈R n×n square, [149,0.3.5]Yet for A∈ R m×n and B ∈ R n×m [55, p.72]det(AB) = det(BA) (1272)det(AB) = detA detB (1273)det(I + AB) = det(I + BA) (1274)For A,B ∈ S n , product AB is symmetric if and only if AB iscommutative;(AB) T = AB ⇔ AB = BA (1275)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.[247, p.26]