v2009.01.01 - Convex Optimization

168 CHAPTER 2. CONVEX GEOMETRY
2.13.7.1 Pointed cones and biorthogonality
Biorthogonality condition X † X = I from Example 2.13.7.0.1 means Γ 1 and
Γ 2 are linearly independent generators of K (B.1.1.1); generators because
every x∈ K is their conic combination. From2.10.2 we know that means
Γ 1 and Γ 2 must be extreme directions of K .
A biorthogonal expansion is necessarily associated with a pointed closed
convex cone; pointed, otherwise there can be no extreme directions (2.8.1).
We will address biorthogonal expansion with respect to a pointed polyhedral
cone having empty interior in2.13.8.
2.13.7.1.1 Example. Expansions implied by diagonalization.
(confer6.5.3.2.1) When matrix X ∈ R M×M is diagonalizable (A.5),
X = SΛS −1 = [ s 1 · · · s M ] Λ⎣
⎡
w T1
.
w T M
⎤
⎦ =
M∑
λ i s i wi T (1438)
coordinates for biorthogonal expansion are its eigenvalues λ i (contained in
diagonal matrix Λ) when expanded in S ;
⎡
X = SS −1 X = [ s 1 · · · s M ] ⎣
i=1
⎤
w1 T X
. ⎦ =
wM TX M∑
λ i s i wi T (352)
Coordinate value depend upon the geometric relationship of X to its linearly
independent eigenmatrices s i w T i . (A.5.1,B.1.1)
Eigenmatrices s i w T i are linearly independent dyads constituted by right
and left eigenvectors of diagonalizable X and are generators of some
pointed polyhedral cone K in a subspace of R M×M .
When S is real and X belongs to that polyhedral cone K , for example,
then coordinates of expansion (the eigenvalues λ i ) must be nonnegative.
When X = QΛQ T is symmetric, coordinates for biorthogonal expansion
are its eigenvalues when expanded in Q ; id est, for X ∈ S M
i=1
X = QQ T X =
M∑
q i qi T X =
i=1
M∑
λ i q i qi T ∈ S M (353)
i=1