v2010.10.26 - Convex Optimization

B.5. ORTHOGONAL MATRIX 647Given X ∈ R n×N , the choice b = 1 N 1 (V S = V ) minimizes ‖X(I − b1 T )‖ F .[164,3.2.1]B.5 Orthogonal matrixB.5.1Vector rotationThe property Q −1 = Q T completely defines an orthogonal matrix Q∈ R n×nemployed to effect vector rotation; [331,2.6,3.4] [333,6.5] [202,2.1]for any x ∈ R n ‖Qx‖ 2 = ‖x‖ 2 (1660)In other words, the 2-norm is orthogonally invariant. A unitary matrix is acomplex generalization of orthogonal matrix; conjugate transpose defines it:U −1 = U H . An orthogonal matrix is simply a real unitary matrix.Orthogonal matrix Q is a normal matrix further characterized:Q −1 = Q T , ‖Q‖ 2 = 1 (1661)Applying characterization (1661) to Q T we see it too is an orthogonal matrix.Hence the rows and columns of Q respectively form an orthonormal set.Normalcy guarantees diagonalization (A.5.1) so, for Q SΛS HSΛ −1 S H = S ∗ ΛS T , ‖δ(Λ)‖ ∞ = 1 (1662)characterizes an orthogonal matrix in terms of eigenvalues and eigenvectors.All permutation matrices Ξ , for example, are nonnegative orthogonalmatrices; and vice versa. Any product of permutation matrices remainsa permutator. Any product of a permutation matrix with an orthogonalmatrix remains orthogonal. In fact, any product of orthogonal matrices AQremains orthogonal by definition. Given any other dimensionally compatibleorthogonal matrix U , the mapping g(A)= U T AQ is a bijection on thedomain of orthogonal matrices (a nonconvex manifold of dimension 1 n(n −1)2[52]). [239,2.1] [240]The largest magnitude entry of an orthogonal matrix is 1; for each andevery j ∈1... n‖Q(j,:)‖ ∞ ≤ 1‖Q(:, j)‖ ∞ ≤ 1(1663)