v2007.09.17 - Convex Optimization

488 APPENDIX A. LINEAR ALGEBRA(AB) T ≠ AB =⎡⎢⎣13 12 −8 −419 25 5 1−5 1 22 9−5 0 9 17⎤⎥⎦ ,⎡λ(AB) = ⎢⎣36.29.10.0.72⎤⎥⎦ (1241)1(AB + 2 (AB)T ) =⎡⎢⎣13 15.5 −6.5 −4.515.5 25 3 0.5−6.5 3 22 9−4.5 0.5 9 17⎤⎥⎦ , λ( 12 (AB + (AB)T ) ) =⎡⎢⎣36.30.10.0.014(1242)Whenever A∈ S n + and B ∈ S n + , then λ(AB)=λ( √ AB √ A) will alwaysbe a nonnegative vector by (1268) and Corollary A.3.1.0.5. Yet positivedefiniteness of the product AB is certified instead by the nonnegativeeigenvalues λ ( 12 (AB + (AB)T ) ) in (1242) (A.3.1.0.1) despite the fact ABis not symmetric. A.6 Horn & Johnson and Zhang resolve the anomaly bychoosing to exclude nonsymmetric matrices and products; they do so byexpanding the domain of test to C n .⎤⎥⎦A.3 Proper statementsof positive semidefinitenessUnlike Horn & Johnson and Zhang, we never adopt the complex domainof test in regard to real matrices. So motivated is our consideration ofproper statements of positive semidefiniteness under real domain of test.This restriction, ironically, complicates the facts when compared to thecorresponding statements for the complex case (found elsewhere, [150] [301]).We state several fundamental facts regarding positive semidefiniteness ofreal matrix A and the product AB and sum A +B of real matrices underfundamental real test (1234); a few require proof as they depart from thestandard texts, while those remaining are well established or obvious.A.6 It is a little more difficult to find a counter-example in R 2×2 or R 3×3 ; which mayhave served to advance any confusion.