v2007.09.13 - Convex Optimization

48 CHAPTER 2. CONVEX GEOMETRY2.2.1.0.2 Definition. Isometric isomorphism.An isometric isomorphism of a vector space having a metric defined on it is alinear bijective mapping T that preserves distance; id est, for all x,y ∈dom T‖Tx − Ty‖ = ‖x − y‖ (39)Then the isometric isomorphism T is a bijective isometry.△Unitary linear operator Q : R n → R n representing orthogonal matrixQ∈ R n×n (B.5), for example, is an isometric ⎡ isomorphism. ⎤ Yet isometric1 0operator T : R 2 → R 3 representing T = ⎣ 0 1 ⎦ on R 2 is injective but not0 0a surjective map [165,1.6] to R 3 .The Frobenius norm is orthogonally invariant; meaning, for X,Y ∈ R p×kand dimensionally compatible orthonormal matrix 2.13 U and orthogonalmatrix Q‖U(X −Y )Q‖ F = ‖X −Y ‖ F (40)2.2.2 Symmetric matrices2.2.2.0.1 Definition. Symmetric matrix subspace.Define a subspace of R M×M : the convex set of all symmetric M×M matrices;S M ∆ = { A∈ R M×M | A=A T} ⊆ R M×M (41)This subspace comprising symmetric matrices S M is isomorphic with thevector space R M(M+1)/2 whose dimension is the number of free variables in asymmetric M ×M matrix. The orthogonal complement [247] [181] of S M isS M⊥ ∆ = { A∈ R M×M | A=−A T} ⊂ R M×M (42)the subspace of antisymmetric matrices in R M×M ; id est,S M ⊕ S M⊥ = R M×M (43)where unique vector sum ⊕ is defined on page 674.△2.13 Any matrix U whose columns are orthonormal with respect to each other (U T U = I);these include the orthogonal matrices.