v2009.01.01 - Convex Optimization

550 APPENDIX A. LINEAR ALGEBRA
Because R(A T A)= R(A T ) and R(AA T )= R(A) , for any A∈ R m×n
rank(AA T ) = rank(A T A) = rankA = rankA T (1372)
For A∈ R m×n having no nullspace, and for any B ∈ R n×k
rank(AB) = rank(B) (1373)
Proof. For any compatible matrix C , N(CAB)⊇ N(AB)⊇ N(B)
is obvious. By assumption ∃A † A † A = I . Let C = A † , then
N(AB)= N(B) and the stated result follows by conservation of
dimension (1469).
For A∈ S n and any nonsingular matrix Y
inertia(A) = inertia(YAY T ) (1374)
a.k.a, Sylvester’s law of inertia. (1415) [96,2.4.3]
For A,B∈R n×n square, [176,0.3.5]
Yet for A∈ R m×n and B ∈ R n×m [68, p.72]
det(AB) = det(BA) (1375)
det(AB) = detA detB (1376)
det(I + AB) = det(I + BA) (1377)
For A,B ∈ S n , product AB is symmetric if and only if AB is
commutative;
(AB) T = AB ⇔ AB = BA (1378)
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.
[287, p.26]