v2009.01.01 - Convex Optimization

556 APPENDIX A. LINEAR ALGEBRA
We can deduce from these, given nonsingular matrix Z and any particular
dimensionally
[ ]
compatible Y : matrix A∈ S M is positive semidefinite if and
Z
T
only if
Y T A[Z Y ] is positive semidefinite. In other words, from the
Corollary it follows: for dimensionally compatible Z
A ≽ 0 ⇔ Z T AZ ≽ 0 and Z T has a left inverse
Products such as Z † Z and ZZ † are symmetric and positive semidefinite
although, given A ≽ 0, Z † AZ and ZAZ † are neither necessarily symmetric
or positive semidefinite.
A.3.1.0.6 Theorem. Symmetric projector semidefinite. [19,III]
[20,6] [193, p.55] For symmetric idempotent matrices P and R
P,R ≽ 0
P ≽ R ⇔ R(P ) ⊇ R(R) ⇔ N(P ) ⊆ N(R)
(1408)
Projector P is never positive definite [289,6.5, prob.20] unless it is the
identity matrix.
⋄
A.3.1.0.7 Theorem. Symmetric positive semidefinite.
Given real matrix Ψ with rank Ψ = 1
Ψ ≽ 0 ⇔ Ψ = uu T (1409)
where u is some real vector; id est, symmetry is necessary and sufficient for
positive semidefiniteness of a rank-1 matrix.
⋄
Proof. Any rank-one matrix must have the form Ψ = uv T . (B.1)
Suppose Ψ is symmetric; id est, v = u . For all y ∈ R M , y T uu T y ≥ 0.
Conversely, suppose uv T is positive semidefinite. We know that can hold if
and only if uv T + vu T ≽ 0 ⇔ for all normalized y ∈ R M , 2y T uv T y ≥ 0 ;
but that is possible only if v = u .
The same does not hold true for matrices of higher rank, as Example A.2.1.0.1
shows.