v2010.10.26 - Convex Optimization

56 CHAPTER 2. CONVEX GEOMETRYConcurrently, consider injective linear operator Py=A † y : R m → R nwhere R(A † )= R(A T ). P(Ax)= PTx achieves projection of vector x onthe row space R(A T ). (E.3.1) This means vector Ax can be succinctlyinterpreted as coefficients of orthogonal projection.Pseudoinverse matrix A † is skinny and full-rank, so operator Py is a linearbijection with respect to its range R(A † ). By Definition 2.2.1.0.1, imageP(T B) of projection PT(B) on R(A T ) in R n must therefore be isomorphicwith the set of projection coefficients T B = {Ax |x∈ B} in R m and have thesame affine dimension by (49). To illustrate, we present a three-dimensionalEuclidean body B in Figure 19 where any point x in the nullspace N(A)maps to the origin.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 (50)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 [331] [250] of S M isS M⊥ { A∈ R M×M | A=−A T} ⊂ R M×M (51)the subspace of antisymmetric matrices in R M×M ; id est,S M ⊕ S M⊥ = R M×M (52)where unique vector sum ⊕ is defined on page 772.△All antisymmetric matrices are hollow by definition (have 0 maindiagonal). Any square matrix A∈ R M×M can be written as a sum of itssymmetric and antisymmetric parts: respectively,A = 1 2 (A +AT ) + 1 2 (A −AT ) (53)The symmetric part is orthogonal in R M2 to the antisymmetric part; videlicet,tr ( (A +A T )(A −A T ) ) = 0 (54)