v2010.10.26 - Convex Optimization

A.3. PROPER STATEMENTS 599(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⎤⎥⎦ (1438)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(1439)Whenever A∈ S n + and B ∈ S n + , then λ(AB)=λ( √ AB √ A) will alwaysbe a nonnegative vector by (1463) and Corollary A.3.1.0.5. Yet positivedefiniteness of product AB is certified instead by the nonnegative eigenvaluesλ ( 12 (AB + (AB)T ) ) in (1439) (A.3.1.0.1) despite the fact that AB is notsymmetric. A.6 Horn & Johnson and Zhang resolve the anomaly by choosingto exclude nonsymmetric matrices and products; they do so by expandingthe domain of test to C n .⎤⎥⎦A.3 Proper statementsof positive semidefinitenessUnlike Horn & Johnson and Zhang, we never adopt a complex domain of testwith real matrices. So motivated is our consideration of proper statementsof positive semidefiniteness under real domain of test. This restriction,ironically, complicates the facts when compared to corresponding statementsfor the complex case (found elsewhere [202] [393]).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 (1431); 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 counterexample in R 2×2 or R 3×3 ; which may haveserved to advance any confusion.