v2004.06.19 - Convex Optimization

2.1. CONVEX SET 272.1.1.1 Frobenius’When Z =Y ∈ R p×k in (19), Frobenius’ norm is resultant;‖Y ‖ 2 F = ‖ vec Y ‖2 2 = 〈Y , Y 〉 = tr(Y T Y ) = δ(Y Y T ) T 1= ∑ i,jY 2ij = ∑ iλ(Y T Y ) i = ∑ iσ(Y ) 2 i(25)where λ(Y T Y ) i is the i th eigenvalue of Y T Y , and σ(Y ) i the i th singularvalue of Y . Were Y a normal matrix (§A.5.2), then σ(Y ) = |λ(Y )|[45, §8.1] thus‖Y ‖ 2 F = ∑ λ(Y ) 2 i = ‖λ(Y )‖ 2 2 (26)iThe converse (26) ⇒ normal Y also holds. [28, §2.5.4] Because the metrics‖ vec X −vec Y ‖ 2 = ‖X −Y ‖ F (27)are equivalent and because vectorization (18) is a linear bijective map, thenvector space R p×k is isometrically isomorphic with vector space R pk in theEuclidean sense and vec is an isometric isomorphism on R p×k (but not onthe vector space that is the range or nullspace associated with a particularmatrix (§2.4, e.g., p.217)). 2.9 Because of this Euclidean structure, all theknown results from convex analysis in Euclidean space R n carry over directlyto the space of real matrices R p×k .The Frobenius norm is orthogonally invariant; meaning, for X,Y ∈ R p×kand compatible orthonormal matrix U and orthogonal matrix Q ,‖U(X −Y )Q‖ F = ‖X −Y ‖ F (28)2.9 An isometric isomorphism of a vector space is a linear bijective mapping T (one-to-oneand onto [38, App.A1.2]) that preserves distance; id est, for all x,y ∈dom T ,‖Tx − Ty‖ = ‖x − y‖Unitary linear operator Q : R n → R n representing orthogonal matrix Q∈ R n×n (§B.5), forexample, is an isometric isomorphism. Yet isometric operator T : R 2 → R 3 representing⎡ ⎤1 0T = ⎣ 0 1 ⎦0 0is injective on R 2 but not a surjective map [38, §1.6] to R 3 .