v2007.09.13 - Convex Optimization

526 APPENDIX B. SIMPLE MATRICESDue 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 (1423)For example, the permutation matrix⎡ ⎤1 0 0Ξ = ⎣ 0 0 1 ⎦ (1424)0 1 0is a Householder matrix having u=[ 0 1 −1 ] T / √ 2 . Not all permutationmatrices are Householder matrices, although all permutation matrices areorthogonal matrices [247,3.4] because they are made by permuting rowsand columns of the identity matrix. Neither are all symmetric permutationmatrices Householder matrices; e.g., Ξ = ⎢⎣Householder matrix.B.4 Auxiliary V -matricesB.4.1Auxiliary projector matrix V⎡0 0 0 10 0 1 00 1 0 01 0 0 0⎤⎥⎦(1502) is not aIt 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 (786) whereV ∆ = I − 1 N 11T ∈ S N (728)is 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 (1425)