v2010.10.26 - Convex Optimization

650 APPENDIX B. SIMPLE MATRICESConsider rotation of these vertices via orthogonal matrixQ [1 1 √ n+1ΞV W ]Ξ ∈ R n+1×n+1 (1672)where permutation matrix Ξ∈ S n+1 is defined in (1728), and V W ∈ R n+1×nis the orthonormal auxiliary matrix defined inB.4.3. This particularorthogonal matrix is selected because it rotates any point in subspace R nabout one axis of revolution onto R ; e.g., rotation Qe n+1 aligns the laststandard basis vector with subspace normal R ⊥ = 1. The rotated standardbasis vectors remaining are orthonormal spanning R .Another interpretation of product QX is rotation/reflection of R(X).Rotation of X as in QXQ T is a simultaneous rotation/reflection of rangeand rowspace. B.8Proof. Any matrix can be expressed as a singular value decompositionX = UΣW T (1563) where δ 2 (Σ) = Σ , R(U) ⊇ R(X) , and R(W) ⊇ R(X T ).B.5.4Matrix rotationOrthogonal matrices are also employed to rotate/reflect other matrices likevectors: [159,12.4.1] Given orthogonal matrix Q , the product Q T A willrotate A∈ R n×n in the Euclidean sense in R n2 because Frobenius’ norm isorthogonally invariant (2.2.1);‖Q T A‖ F = √ tr(A T QQ T A) = ‖A‖ F (1673)(likewise for AQ). Were A symmetric, such a rotation would depart from S n .One remedy is to instead form product Q T AQ because‖Q T AQ‖ F = √ tr(Q T A T QQ T AQ) = ‖A‖ F (1674)ByA.1.1 no.31,vec Q T AQ = (Q ⊗ Q) T vec A (1675)B.8 The product Q T AQ can be regarded as a coordinate transformation; e.g., givenlinear map y =Ax : R n → R n and orthogonal Q, the transformation Qy =AQx is arotation/reflection of range and rowspace (141) of matrix A where Qy ∈ R(A) andQx∈ R(A T ) (142).