v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 169
becomes an orthogonal expansion with orthonormality condition Q T Q=I
where λ i is the i th eigenvalue of X , q i is the corresponding i th eigenvector
arranged columnar in orthogonal matrix
Q = [q 1 q 2 · · · q M ] ∈ R M×M (354)
and where eigenmatrix q i qi
T is an extreme direction of some pointed
polyhedral cone K ⊂ S M and an extreme direction of the positive semidefinite
cone S M + .
Orthogonal expansion is a special case of biorthogonal expansion of
X ∈ aff K occurring when polyhedral cone K is any rotation about the
origin of an orthant belonging to a subspace.
Similarly, when X = QΛQ T belongs to the positive semidefinite cone in
the subspace of symmetric matrices, coordinates for orthogonal expansion
must be its nonnegative eigenvalues (1353) when expanded in Q ; id est, for
X ∈ S M +
M∑
M∑
X = QQ T X = q i qi T X = λ i q i qi T ∈ S M + (355)
i=1
where λ i ≥0 is the i th eigenvalue of X . This means X simultaneously
belongs to the positive semidefinite cone and to the pointed polyhedral cone
K formed by the conic hull of its eigenmatrices.
2.13.7.1.2 Example. Expansion respecting nonpositive orthant.
Suppose x ∈ K any orthant in R n . 2.62 Then coordinates for biorthogonal
expansion of x must be nonnegative; in fact, absolute value of the Cartesian
coordinates.
Suppose, in particular, x belongs to the nonpositive orthant K = R n − .
Then the biorthogonal expansion becomes an orthogonal expansion
n∑
n∑
x = XX T x = −e i (−e T i x) = −e i |e T i x| ∈ R n − (356)
i=1
and the coordinates of expansion are nonnegative. For this orthant K we have
orthonormality condition X T X = I where X = −I , e i ∈ R n is a standard
basis vector, and −e i is an extreme direction (2.8.1) of K .
Of course, this expansion x=XX T x applies more broadly to domain R n ,
but then the coordinates each belong to all of R .
2.62 An orthant is simplicial and self-dual.
i=1
i=1