v2010.10.26 - Convex Optimization

640 APPENDIX B. SIMPLE MATRICESN(u T )R(E) = N(v T )0 0 N(E)=R(u)R(v)R N = N(u T ) ⊞ N(E)R(v) ⊞ R(E) = R NFigure 158: v T u = 1/ζ . The four fundamental subspaces)[333,3.6] ofelementary matrix E as a linear mapping E(x)=(I − uvT x .v T uwhere λ 1 > 0 >λ 2 , with corresponding eigenvectorsx 1 u‖u‖ + v‖v‖ ,x 2 u‖u‖ − v‖v‖(1634)spanning the doublet range. Eigenvalue λ 1 cannot be 0 unless u and v haveopposing directions, but that is antithetical since then the dyads would nolonger be independent. Eigenvalue λ 2 is 0 if and only if u and v share thesame direction, again antithetical. Generally we have λ 1 > 0 and λ 2 < 0, soΠ is indefinite.By the nullspace and range of dyad sum theorem, doublet Π hasN −2 ([ zero-eigenvalues ]) remaining and corresponding eigenvectors spanningvTNu T . We therefore haveR(Π) = R([u v ]) , N(Π) = v ⊥ ∩ u ⊥ (1635)of respective dimension 2 and N −2.B.3 Elementary matrixA matrix of the formE = I − ζuv T ∈ R N×N (1636)where ζ ∈ R is finite and u,v ∈ R N , is called an elementary matrix or arank-one modification of the identity. [204] Any elementary matrix in R N×N