ABSTRACT
This chapter presents the results from Hendriks and Landsman (1998) [154].
LetMatp,r be the vector space of real p×r,r≤ pmatrices, with the inner product (A,B) = TrATB, so that ‖A−B‖2 = Tr(A−B)T (A−B). This corresponds to the Euclidean inner product with respect to the identification of Matp,r with
Rpr by putting the matrix entries in one column of size pr. Let Sym(r) be the vector space of symmetric r× r matrices, and Sym+r the subset of semi positive-definite symmetric r× r matrices. Let Er denote the r× r unit matrix. Then M = Vp,r ⊂ Matp,r denotes the Stiefel manifold M = Vp,r = {V ∈ Matp,r,V
TV = Er ∈ Sym(r)} whose dimension is m = pr− r(r+1)2 . M can be considered as a compact submanifold of the vector space Matp,r given by the
equation
f (X) = XTX = Er, (9.1)
where f is considered as a mapping between vector spaces with inner product
f :Matp,r → Sym(r). Therefore, a point in the Stiefel manifold is a p× r matrix with orthogonal columns of Euclidean norm 1. As a particular case Vk,1 corresponds to the sphere Sk−1, and Vp,p equals the orthogonal group O(p) of p× p-matrices X such that XTX = Ep. Recall that the Stiefel manifoldM = Vp,r is a homogeneous space with re-
spect to the Lie group O(p) where the action is defined as the restriction of the
action by matrix multiplication
O(p)×Matp,r (g,X) 7→ gX ∈Matp,r. (9.2)
As a matter of fact,M admits the larger symmetry group O(p)×O(r), where
the action is defined by restriction of the action
(O(p)×O(r))×Matp,r (g,h,X) 7→ gXhT ∈Matp,r. (9.3)
It is clear that these symmetries are in fact isometries. Namely, it can be shown
that ‖g(x− y)hT‖2 = ‖(x− y)‖2. If U ∈Matp,r, then the elements X ofM for which LU (X) = ‖U−X‖2 = (U−X ,U−X) is minimal are characterized by the condition thatU =Xc, where c∈ Sym+r is the (semi) positive-definite symmetric square root of UTU . X is uniquely defined if c is non singular, thus if U is
of rank r. In that case, X is a non degenerate critical point of the function
LU :M→ R. If U is of rank less than r, uniqueness of X fails as there is a 1-1 correspondence between isometric (injective) mappings Y : Ker U →(Im U)⊥ and solutionsV to the equationU =Vc, given by Y =V |kerU . As such, the cut-locus C consists exactly of the p× r-matrices U that are not of maximal rank, r. The nearest-point mapping Φ :Matp,r \C →M will be given by
Φ(U) =U(UTU)−1/2,U /∈ C (9.4)
Notice that the linear mapping Df |X : Matp,r → Sym(r) is given by Df |X (H) =HTX +XTH and that its transpose (or rather adjoint with respect to the inner products) (Df |X )T : Sym(r)→Matp,r is given by (Df |X )T (c) = 2Xc. Thus, for µ ∈M, we have Df |µ (Df |µ )T (c) = 4c. The orthogonal projection on the tangent space Tµ (M) will be given by
tanµ(U) =U− (Df |µ)T (Df |µ (Df |µ )T )−1Df |µ (U) =U − 1 2
µ[µTU +UTµ].
