v2010.10.26 - Convex Optimization

642 APPENDIX B. SIMPLE MATRICESwhich is a symmetric orthogonal (reflection) matrix (H −1 =H T =H(B.5.2)). Vector u is normal to an N−1-dimensional subspace u ⊥ throughwhich this particular H effects pointwise reflection; e.g., Hu ⊥ = u ⊥ whileHu = −u .Matrix H has N −1 orthonormal eigenvectors spanning that reflectingsubspace u ⊥ with corresponding eigenvalues equal to 1. The remainingeigenvector u has corresponding eigenvalue −1 ; sodetH = −1 (1643)Due to symmetry of H , the matrix 2-norm (the spectral norm) is equal to thelargest eigenvalue-magnitude. A Householder matrix is thus characterized,H T = H , H −1 = H T , ‖H‖ 2 = 1, H 0 (1644)For example, the permutation matrix⎡ ⎤1 0 0Ξ = ⎣ 0 0 1 ⎦ (1645)0 1 0is a Householder matrix having u=[ 0 1 −1 ] T / √ 2 . Not all permutationmatrices are Householder matrices, although all permutation matricesare orthogonal matrices (B.5.1, Ξ T Ξ = I) [331,3.4] because they aremade by permuting rows and columns of the identity matrix. Neitherare ⎡all symmetric ⎤ permutation matrices Householder matrices; e.g.,0 0 0 1Ξ = ⎢ 0 0 1 0⎥⎣ 0 1 0 0 ⎦ (1728) is not a Householder matrix.1 0 0 0B.4 Auxiliary V -matricesB.4.1Auxiliary projector matrix VIt is convenient to define a matrix V that arises naturally as a consequence oftranslating the geometric center α c (5.5.1.0.1) of some list X to the origin.In place of X − α c 1 T we may write XV as in (974) whereV = I − 1 N 11T ∈ S N (913)