v2009.01.01 - Convex Optimization

A.7. ZEROS 575
A.7.5.0.1 Proposition. (Sturm) Dyad-decompositions. [292,5.2]
Let positive semidefinite matrix X ∈ S M + have rank ρ . Then given symmetric
matrix A∈ S M , 〈A, X〉 = 0 if and only if there exists a dyad-decomposition
satisfying
X =
ρ∑
x j xj T (1488)
j=1
〈A , x j x T j 〉 = 0 for each and every j ∈ {1... ρ} (1489)
⋄
The dyad-decomposition of X proposed is generally not that obtained
from a standard diagonalization by eigen decomposition, unless ρ =1 or
the given matrix A is diagonalizable simultaneously (A.7.4) with X .
That means, elemental dyads in decomposition (1488) constitute a generally
nonorthogonal set. Sturm & Zhang give a simple procedure for constructing
the dyad-decomposition (Wıκımization); matrix A may be regarded as a
parameter.
A.7.5.0.2 Example. Dyad.
The dyad uv T ∈ R M×M (B.1) is zero definite on all x for which either
x T u=0 or x T v=0;
{x | x T uv T x = 0} = {x | x T u=0} ∪ {x | v T x=0} (1490)
id est, on u ⊥ ∪ v ⊥ . Symmetrizing the dyad does not change the outcome:
{x | x T (uv T + vu T )x/2 = 0} = {x | x T u=0} ∪ {x | v T x=0} (1491)