v2009.01.01 - Convex Optimization

A.3. PROPER STATEMENTS 545
(AB) T ≠ AB =
⎡
⎢
⎣
13 12 −8 −4
19 25 5 1
−5 1 22 9
−5 0 9 17
⎤
⎥
⎦ ,
⎡
λ(AB) = ⎢
⎣
36.
29.
10.
0.72
⎤
⎥
⎦ (1341)
1
(AB + 2 (AB)T ) =
⎡
⎢
⎣
13 15.5 −6.5 −4.5
15.5 25 3 0.5
−6.5 3 22 9
−4.5 0.5 9 17
⎤
⎥
⎦ , λ( 1
2 (AB + (AB)T ) ) =
⎡
⎢
⎣
36.
30.
10.
0.014
(1342)
Whenever A∈ S n + and B ∈ S n + , then λ(AB)=λ( √ AB √ A) will always
be a nonnegative vector by (1367) and Corollary A.3.1.0.5. Yet positive
definiteness of the product AB is certified instead by the nonnegative
eigenvalues λ ( 1
2 (AB + (AB)T ) ) in (1342) (A.3.1.0.1) despite the fact AB
is not symmetric. A.6 Horn & Johnson and Zhang resolve the anomaly by
choosing to exclude nonsymmetric matrices and products; they do so by
expanding the domain of test to C n .
⎤
⎥
⎦
A.3 Proper statements
of positive semidefiniteness
Unlike Horn & Johnson and Zhang, we never adopt a complex domain of test
with real matrices. So motivated is our consideration of proper statements
of positive semidefiniteness under real domain of test. This restriction,
ironically, complicates the facts when compared to the corresponding
statements for the complex case (found elsewhere, [176] [344]).
We state several fundamental facts regarding positive semidefiniteness of
real matrix A and the product AB and sum A +B of real matrices under
fundamental real test (1334); a few require proof as they depart from the
standard 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 may
have served to advance any confusion.