v2007.09.13 - Convex Optimization

B.5. ORTHOGONAL MATRIX 533B.5.3.0.1 Example. One axis of revolution. [Partition an n+1-dimensional Euclidean space R n+1 =∆ RnRn-dimensional subspace]and define anR ∆ = {λ∈ R n+1 | 1 T λ = 0} (1446)(a hyperplane through the origin). We want an orthogonal matrix thatrotates a list in the columns of matrix X ∈ R n+1×N through the dihedralangle between R n and R (2.4.3)( ) ( )(R n 〈en+1 , 1〉1, R) = arccos = arccos √ radians (1447)‖e n+1 ‖ ‖1‖ n+1The vertex-description of the nonnegative orthant in R n+1 is{[e 1 e 2 · · · e n+1 ]a | a ≽ 0} = {a ≽ 0} = R n+1+ ⊂ R n+1 (1448)Consider rotation of these vertices via orthogonal matrixQ ∆ = [1 1 √ n+1ΞV W ]Ξ ∈ R n+1×n+1 (1449)where permutation matrix Ξ∈ S n+1 is defined in (1502), 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 the simultaneous rotation/reflection of rangeand rowspace. B.8Proof. Any matrix can be expressed as a singular value decompositionX = UΣW T (1345) where δ 2 (Σ) = Σ , R(U) ⊇ R(X) , and R(W) ⊇ R(X T ).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 the range and rowspace (119) of matrix A where Qy ∈ R(A) andQx∈ R(A T ) (120).