v2007.11.26 - Convex Optimization

540 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 120: v T u = 1/ζ . The four fundamental subspaces)[253,3.6] ofelementary matrix E as a linear mapping E(x)=(I − uvT x .v T uBy 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 ⊥ (1445)of respective dimension 2 and N −2.B.3 Elementary matrixA matrix of the formE = I − ζuv T ∈ R N×N (1446)where ζ ∈ R is finite and u,v ∈ R N , is called an elementary matrix or arank-one modification of the identity. [152] Any elementary matrix in R N×Nhas N −1 eigenvalues equal to 1 corresponding to real eigenvectors thatspan v ⊥ . The remaining eigenvalueλ = 1 − ζv T u (1447)corresponds to eigenvector u . B.6 From [162, App.7.A.26] the determinant:detE = 1 − tr ( ζuv T) = λ (1448)B.6 Elementary matrix E is not always diagonalizable because eigenvector u need not beindependent of the others; id est, u∈v ⊥ is possible.