v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization 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)
B.4. AUXILIARY V -MATRICES 527the 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 (1426)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 (1427)The {0, 1} eigenvalues also indicate diagonalizable V is a projectionmatrix. [298,4.1, thm.4.1] Symmetry of V denotes orthogonal projection;from (1678),V T = V , V † = V , ‖V ‖ 2 = 1, V ≽ 0 (1428)Matrix V is also circulant [117].B.4.1.0.1 Example. Relationship of auxiliary to Householder matrix.Let H ∈ S N be a Householder matrix (1421) defined by⎡ ⎤u = ⎢1.⎥⎣ 11 + √ ⎦ ∈ RN (1429)NThen we have [105,2]Let D ∈ S N h and define[ I 0V = H0 T 0[−HDH = ∆ A b−b T c]H (1430)](1431)where b is a vector. Then because H is nonsingular (A.3.1.0.5) [132,3][ ] A 0−V DV = −H0 T H ≽ 0 ⇔ −A ≽ 0 (1432)0and affine dimension is r = rankA when D is a Euclidean distance matrix.
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B.4. AUXILIARY V -MATRICES 527the 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 (1426)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 (1427)The {0, 1} eigenvalues also indicate diagonalizable V is a projectionmatrix. [298,4.1, thm.4.1] Symmetry of V denotes orthogonal projection;from (1678),V T = V , V † = V , ‖V ‖ 2 = 1, V ≽ 0 (1428)Matrix V is also circulant [117].B.4.1.0.1 Example. Relationship of auxiliary to Householder matrix.Let H ∈ S N be a Householder matrix (1421) defined by⎡ ⎤u = ⎢1.⎥⎣ 11 + √ ⎦ ∈ RN (1429)NThen we have [105,2]Let D ∈ S N h and define[ I 0V = H0 T 0[−HDH = ∆ A b−b T c]H (1430)](1431)where b is a vector. Then because H is nonsingular (A.3.1.0.5) [132,3][ ] A 0−V DV = −H0 T H ≽ 0 ⇔ −A ≽ 0 (1432)0and affine dimension is r = rankA when D is a Euclidean distance matrix.
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