v2007.09.17 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 1132.9.2.5.2 Example. PSD cone inscription in three dimensions.Theorem. Geršgorin discs. [150,6.1] [274]For p∈R m + given A=[A ij ]∈ S m , then all eigenvalues of A belong to the unionof m closed intervals on the real line;λ(A) ∈ m ⋃i=1⎧⎪⎨ξ ∈ R⎪⎩|ξ − A ii | ≤ ̺i∆= 1 m∑p ij=1j ≠ i⎫⎪⎬⋃p j |A ij | = m [A ii −̺i , A ii +̺i]i=1 ⎪⎭(206)Furthermore, if a union of k of these m [intervals] forms a connected regionthat is disjoint from all the remaining n −k [intervals], then there areprecisely k eigenvalues of A in this region.⋄To apply the theorem to determine positive semidefiniteness of symmetricmatrix A , we observe that for each i we must haveSupposeA ii ≥ ̺i (207)m = 2 (208)so A ∈ S 2 . Vectorizing A as in (47), svec A belongs to isometricallyisomorphic R 3 . Then we have m2 m−1 = 4 inequalities, in the matrix entriesA ij with Geršgorin parameters p =[p i ]∈ R 2 + ,p 1 A 11 ≥ ±p 2 A 12p 2 A 22 ≥ ±p 1 A 12(209)which describe an intersection of four halfspaces in R m(m+1)/2 . Thatintersection creates the polyhedral proper cone K (2.12.1) whoseconstruction is illustrated in Figure 36. Drawn truncated is the boundaryof the positive semidefinite cone svec S 2 + and the bounding hyperplanessupporting K .Created by means of Geršgorin discs, K always belongs to the positivesemidefinite cone for any nonnegative value of p ∈ R m + . Hence any point inK corresponds to some positive semidefinite matrix A . Only the extremedirections of K intersect the positive semidefinite cone boundary in thisdimension; the four extreme directions of K are extreme directions of the