v2010.10.26 - Convex Optimization

124 CHAPTER 2. CONVEX GEOMETRY2.9.2.5 PSD cone face containing principal submatrixA principal submatrix of a matrix A∈ R M×M is formed by discarding anyparticular subset of its rows and columns having the same indices. Thereare M!/(1!(M −1)!) principal 1 × 1 submatrices, M!/(2!(M −2)!) principal2 × 2 submatrices, and so on, totaling 2 M − 1 principal submatricesincluding A itself. Principal submatrices of a symmetric matrix aresymmetric. A given symmetric matrix has rank ρ iff it has a nonsingularprincipal ρ×ρ submatrix but none larger. [297,5-10] By loading vectory in test y T Ay (A.2) with various binary patterns, it follows thatany principal submatrix must be positive (semi)definite whenever A is(Theorem A.3.1.0.4). If positive semidefinite matrix A∈ S M + has principalsubmatrix of dimension ρ with rank r , then rankA ≤ M −ρ+r by (1536).Because each and every principal submatrix of a positive semidefinitematrix in S M is positive semidefinite, then each principal submatrix belongsto a certain face of positive semidefinite cone S M + by (222). Of specialinterest are full-rank positive semidefinite principal submatrices, for thendescription of smallest face becomes simpler. We can find the smallest face,that contains a particular full-rank principal submatrix of A , by embeddingthat submatrix in a 0 matrix of the same dimension as A : Were Φ a binarydiagonal matrixΦ = δ 2 (Φ)∈ S M , Φ ii ∈ {0, 1} (224)having diagonal entry 0 corresponding to a discarded row and columnfrom A∈ S M + , then any principal submatrix 2.42 so embedded canbe expressed ΦAΦ ; id est, for an embedded principal submatrixΦAΦ∈ S M + rank ΦAΦ = rank Φ ≤ rankAF ( S M + ∋ΦAΦ ) = {X ∈ S M + | N(X) ⊇ N(ΦAΦ)}= {X ∈ S M + | 〈I − Φ , X〉 = 0}= {ΦΨΦ | Ψ ∈ S M + }≃ S rankΦ+(225)Smallest face that contains an embedded principal submatrix, whose rankis not necessarily full, may be expressed like (221): For embedded principal] I 02.42 To express a leading principal submatrix, for example, Φ =[0 T .0