v2010.10.26 - Convex Optimization

B.5. ORTHOGONAL MATRIX 649Figure 159: Gimbal: a mechanism imparting three degrees of dimensionalfreedom to a Euclidean body suspended at the device’s center. Each ring isfree to rotate about one axis. (Courtesy of The MathWorks Inc.)the precise meaning of rotation is best illustrated in Figure 159 where thegimbal aids visualization of what is achievable; mathematically, (5.5.2.0.1)⎡⎤⎡⎤⎡⎤cosθ 0 − sin θ 1 0 0 cos φ − sin φ 0Q = ⎣ 0 1 0 ⎦⎣0 cos ψ − sin ψ ⎦⎣sin φ cos φ 0 ⎦ (1668)sin θ 0 cos θ 0 sinψ cos ψ 0 0 1B.5.3.0.1 Example. One axis of revolution. [ RPartition an n+1-dimensional Euclidean space R n+1 nRan n-dimensional subspace]and defineR {λ∈ R n+1 | 1 T λ = 0} (1669)(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 (1670)‖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 (1671)