v2010.10.26 - Convex Optimization

604 APPENDIX A. LINEAR ALGEBRABecause R(A T A)= R(A T ) and R(AA T )= R(A) (p.634), for anyA∈ R m×n rank(AA T ) = rank(A T A) = rankA = rankA T (1468)For A∈ R m×n having no nullspace, and for any B ∈ R n×krank(AB) = rank(B) (1469)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 (1583).For A∈ S n and any nonsingular matrix Yinertia(A) = inertia(YAY T ) (1470)a.k.a, Sylvester’s law of inertia. (1516) [113,2.4.3]For A,B∈R n×n square, [202,0.3.5]Yet for A∈ R m×n and B ∈ R n×m [77, p.72]det(AB) = det(BA) (1471)det(AB) = detA detB (1472)det(I + AB) = det(I + BA) (1473)For A,B ∈ S n , product AB is symmetric if and only if AB iscommutative;(AB) T = AB ⇔ AB = BA (1474)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.[331, p.26]