Ley 618: Ley General de Higiene y Seguridad

In this section the Stiefel manifold is represented as the collection of all N ×p semi- unitary matrices, viewed as a subset of CN ×p. This is in contrast to the quotient space

Table 1. Horizontal HX0 and Vertical VX0 Space Dimension of Quotient Manifolds

Manifold BY AY HY VY

St(N,p) 2p(N − p) p2 2pN − p2 (N − p)2

B(N,p) 2p(N − p) p2− p 2pN − p2− p (N − p)2+ p

G(N,p) 2p(N − p) 0 2p(N − p) (N − p)2+ p2

representation where points on the manifold were identified with subsets of U (N ) ⊂ CN ×N_{. For this complex case the Stiefel manifold is defined in concrete form as}

St(N,p) ≡ {Y ∈ CN ×p: YHY = I} . (223)

When p = N , St(N,p)is the unitary group U (N ).

Tangent Space

In the quotient manifold approach the ambient space was N2_{-dimensional and tan-}

gent vectors were represented as N × N matrices of the form (171) or (165). Since, in this section, we are working in an N p-dimensional ambient space, here tangent vectors are represented as N × p matrices. For the arbitrary curve Y (t) on St(N,p)differentiating

the defining equation (223) yields

d dt(Y

H_{(t)Y (t)) = ˙Y}H_{Y + Y}H_{Y = 0 .˙ (224)}

Introducing the notation ∆ ≡ ˙Y this becomes

∆HY + YH∆ = 0 . (225)

An arbitrary N × p matrix ∆ ∈ TYCN ×pmay be decomposed as

∆ = Y (A + S) + Y⊥B (226)

for A, p×p anti-Hermitian, S, p×p Hermitian , B, arbitrary (N −p)×p complex matrix and YHY⊥ = 0. The tangent space TYSt(N,p) < TYCN ×pis constrained by tangency

condition (225). Substituting (226) into (225) shows that the symmetric part of (226) must vanish, that is S ≡ 0. Elements of TYSt(N,p) are therefore

∆ = Y A + Y⊥B . (227)

This form is the same as found by selecting the first p-columns of the quotient space tangent vector representation (165). The space TYSt(N,p) partitions as before (176), that

is

TYSt(N,p) = AY ⊕ BY (228)

where AY and BY and defined by selecting the first p-columns in the definitions (172)

and (174). respectively. The symmetric component of (226), Y S, is normal to the em- bedded manifold in the ambient space. The collection of all such vectors is the normal space (TYSt(N,p))⊥. The total space decomposes as

TYCN ×p= TYSt(N,p)⊕ (TYSt(N,p))⊥ (229)

Note that for p = 1, Y is N × 1 unit vector an S is a complex scalar.

Canonical Metric

A metric on manifold is a smoothly varying mapping

g : TYM × TYM 7→ R (230)

that at each point satisfies the standard inner product conditions (see section A.3.2). The standard Euclidean metric on M = CN ×p_{, viewed as a vector space over the reals, is}

ge(∆, ∆) = 2 · Re(tr(∆H∆)) (231)

The scale factor 2 has been included for reasons of simplicity that will become evident in the following. The subscript “e” stands for Euclidean. Substituting the tangent vector ∆ ∈ TYSt(N,p)form given by (227) into (231) yields

ge(∆, ∆) = 2 · Re(tr(BHB) + tr(AHA)) (232)

This is not the metric selected for St(N,p) since it counts independent elements of A of

two times. To see this consider the anti-Hermitian matrix

A = " 0 −a∗ a 0 # . (233) Then trAHA= tr " |a|2 ₀ 0 |a|2 #! = 2|a|2 (234)

We see that the independent element a is counted twice. To correct this, the canonical metric on St(N,p), denoted as gc, is instead selected as

gc(∆, ∆) = 2 · Re  tr(∆H(I −1 2Y Y H_)∆) _{. (235)} or equivalently as gc(∆, ∆) = 2 · Re  tr(∆H(Y⊥Y⊥H + 1 2Y Y H_)∆) _{. (236)}

For an arbitrary ∆ ∈ TYSt(N,p)given by (227) this yields

gc(∆, ∆) = 2 · Re 

tr(BHB) + tr(AHA) . (237)

Comparing the above result (236) to (199), the metric for the quotient manifold formulation, we see that the two forms they are identical. Therefore, the formulas for geodesics for the Stiefel manifold given in section 2.2.4 are correct if we view the Stiefel manifold as the set of semi-unitary N × p matrices with the metric gc(235). The canon-

ical metric gcis thus not simply the restriction of the geto the submanifold.

For simplicity we sometimes will use the alternate angle bracket notation for the metric

h∆, ∆i ≡ gc(∆, ∆) (238)

Geodesics (Stiefel)

Let Y (t) denote an arbitrary smooth curve in St(N,p)between the points Y0and Y1,

such Y0 = Y (0) and Y1 = Y (t1). The length of this curve is given by the integral

L = Z t1 0 r gc  ˙ Y , ˙Ydt (239)

where ˙Y = dY_dt is a tangent to the curve at Y (t).

The Euler-Lagrange equations of motion (geodesic equation) are derived from this path length integral using the Calculus of Variations. The path length defined above is minimized over the set of all smooth curves between Y0and Y1. This minimization leads

a 2nd order nonlinear differential equation called the geodesic equation. For the choice of the canonical metric (236) this equation is derived in [?] as

¨

Y + ˙Y ( ˙YHY ) + Y (YHY )˙ 2+ ˙YHY = 0 .˙ (240)

Direct substitution verifies that general solution is given by

Y (t) =h Y0 Y0⊥ i exp t " A −BH B 0 #! IN,p. (241)

As anticipated, this is identical to the geodesic derived from the quotient manifold for- mula and given by (167).

Basis Manifold as a Quotient manifold of the Stiefel Manifold

The Basis manifold can be naturally defined as a quotient of the above concrete Stiefel manifold formulation. Recalling (207), the Basis manifold is represented as the quotient St(N,p)/Dpwith each point being an equivalence class given by

bY c = {Y M : M ∈ Dp} . (242)

A point in the Basis manifold is thus a particular subset of the N × p semi-unitary matrices, and the Basis manifold itself is the collection of all these subsets. When

performing computations on the Basis manifold we will use an N × p matrix Y to represent an entire equivalence class; any representative of the class will do.

The vertical space, a subspace of TY0St(N,p), corresponds to motions that remain in

the set (242) and consists of tangent vectors of the form

Φ = Y0M ; M ∈ Dp (243)

where D is unitary-diagonal. This collection defines the vertical space V0 at Y0. The

orthogonal complement is the horizontal space, HY0, which here consists of vectors of

the form

∆ = Y0A + Y0⊥B . (244)

The significance of the horizontal space is that it provides a representation of tangents to the quotient manifold space which in the current discussion is BN,p, that is

TY0BN,p≡ HY0 (245)

Movements in the vertical direction make no change in the quotient space. Therefore, the metric and geodesics must all be restricted to the horizontal space. Viewed as a quotient manifold of St(N,p) we have

dim(B(N,p)) = dim(St(N,p))) − dim(Dp) = 2N p − p2− p = 2p(N − p) + p2− p (246)

The Grassmann manifold can be defined as a quotient manifold of St(N,p) in a similar

way.