v2010.10.26 - Convex Optimization

B.4. AUXILIARY V -MATRICES 643is an elementary matrix called the geometric centering matrix.Any elementary matrix in R N×N has N −1 eigenvalues equal to 1. For theparticular elementary matrix V , the N th eigenvalue equals 0. The numberof 0 eigenvalues must equal dim N(V ) = 1, by the 0 eigenvalues theorem(A.7.3.0.1), because V =V T is diagonalizable. BecauseV 1 = 0 (1646)the nullspace N(V )= R(1) is spanned by the eigenvector 1. The remainingeigenvectors span R(V ) ≡ 1 ⊥ = N(1 T ) that has dimension N −1.BecauseV 2 = V (1647)and V T = V , elementary matrix V is also a projection matrix (E.3)projecting orthogonally on its range N(1 T ) which is a hyperplane containingthe origin in R N V = I − 1(1 T 1) −1 1 T (1648)The {0, 1} eigenvalues also indicate diagonalizable V is a projectionmatrix. [393,4.1 thm.4.1] Symmetry of V denotes orthogonal projection;from (1916),V T = V , V † = V , ‖V ‖ 2 = 1, V ≽ 0 (1649)Matrix V is also circulant [168].B.4.1.0.1 Example. Relationship of Auxiliary to Householder matrix.Let H ∈ S N be a Householder matrix (1642) defined by⎡ ⎤u = ⎢1.⎥⎣ 11 + √ ⎦ ∈ RN (1650)NThen we have [155,2]Let D ∈ S N h and define[ I 0V = H0 T 0[ A b−HDH −b T c]H (1651)](1652)