v2010.10.26 - Convex Optimization

B.1. RANK-ONE MATRIX (DYAD) 635R(v)0 0 R(Ψ) = R(u)N(Ψ)= N(v T )N(u T )R N = R(v) ⊞ N(uv T )N(u T ) ⊞ R(uv T ) = R MFigure 156: The four fundamental subspaces [333,3.6] of any dyadΨ = uv T ∈R M×N : R(v)⊥ N(Ψ) & N(u T )⊥ R(Ψ). Ψ(x) uv T x is a linearmapping from R N to R M . Map from R(v) to R(u) is bijective. [331,3.1]When dyad uv T ∈R N×N is square, uv T has at least N −1 0-eigenvaluesand corresponding eigenvectors spanning v ⊥ . The remaining eigenvector uspans the range of uv T with corresponding eigenvalueλ = v T u = tr(uv T ) ∈ R (1612)Determinant is a product of the eigenvalues; so, it is always true thatdet Ψ = det(uv T ) = 0 (1613)When λ = 1, the square dyad is a nonorthogonal projector projecting on itsrange (Ψ 2 =Ψ ,E.6); a projector dyad. It is quite possible that u∈v ⊥making the remaining eigenvalue instead 0 ; B.2 λ = 0 together with thefirst N −1 0-eigenvalues; id est, it is possible uv T were nonzero while allits eigenvalues are 0. The matrix[ ] 1 [ 1 1]=−1[ 1 1−1 −1](1614)for example, has two 0-eigenvalues. In other words, eigenvector u maysimultaneously be a member of the nullspace and range of the dyad.The explanation is, simply, because u and v share the same dimension,dimu = M = dimv = N :B.2 A dyad is not always diagonalizable (A.5) because its eigenvectors are not necessarilyindependent.