v2007.09.13 - Convex Optimization

A.3. PROPER STATEMENTS 495A.3.1.0.2 Theorem. (Weyl) Eigenvalues of sum. [149,4.3]For A,B∈ R M×M , place the eigenvalues of each symmetrized matrix intothe respective vectors λ ( 1(A 2 +AT ) ) , λ ( 1(B 2 +BT ) ) ∈ R M in nonincreasingorder so λ ( 1(A 2 +AT ) ) holds the largest eigenvalue of symmetrized A while1λ ( 1(B 2 +BT ) ) holds the largest eigenvalue of symmetrized B , and so on.1Then, for any k ∈{1... M }λ ( A +A T) k + λ( B +B T) M ≤ λ( (A +A T ) + (B +B T ) ) k ≤ λ( A +A T) k + λ( B +B T) 1(1301)Weyl’s theorem establishes positive semidefiniteness of a sum of positivesemidefinite matrices. Because S M + is a convex cone (2.9.0.0.1), thenby (144)A,B ≽ 0 ⇒ ζA + ξB ≽ 0 for all ζ,ξ ≥ 0 (1302)⋄A.3.1.0.3 Corollary. Eigenvalues of sum and difference. [149,4.3]For A∈ S M and B ∈ S M + , place the eigenvalues of each matrix into therespective vectors λ(A), λ(B)∈ R M in nonincreasing order so λ(A) 1 holdsthe largest eigenvalue of A while λ(B) 1 holds the largest eigenvalue of B ,and so on. Then, for any k ∈{1... M}λ(A − B) k ≤ λ(A) k ≤ λ(A +B) k (1303)⋄When B is rank-one positive semidefinite, the eigenvalues interlace; id est,for B = qq Tλ(A) k−1 ≤ λ(A − qq T ) k ≤ λ(A) k ≤ λ(A + qq T ) k ≤ λ(A) k+1 (1304)