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Complex Stiefel and Grassmann tangent projections

Statement

For $X \in {St}_{C} (n, r)$ , as a real manifold, $T_{X} {St}_{C} (n, r) = {Z : X^{*} Z + Z^{*} X = 0}$ and $Π_{X}^{S t} (G) = G - X Herm (X^{*} G)$ . The complex Grassmann tangent has the unique horizontal representative $X^{*} Z = 0$ , and its quotient-metric projection is $Π_{X}^{G r} (G) = (I - X X^{*}) G$ .

Facts & Assumptions

Given: $1 \leq r \leq n$ , $X^{*} X = I_{r}$ , and the real Frobenius metrics of Complex Stiefel and Grassmann optimization conventions.

[F1]

The Stiefel frame set is the level set of $X \mapsto X^{*} X - I_{r}$ , and the Grassmann space is its quotient by the right unitary action (Complex Stiefel and Grassmann optimization conventions).

[F2]

The tangent of a regular level set is the kernel of the level-map differential (The tangent space of a regular level set is the kernel).

[F3]

The complex Grassmann quotient has graph charts, and its frame bundle has right unitary fibers (Stiefel spaces, Grassmannians, and tautological bundles).

Proof

technique · differentiate the Hermitian constraint and orthogonally decompose into vertical, horizontal, and normal components

1.1F1F2givenalgebra

Regard the Hermitian $r \times r$ matrices as a real vector space. The derivative of $F (X) = X^{*} X - I_{r}$ is $D F_{X} [Z] = X^{*} Z + Z^{*} X$ . It is onto that space: for any Hermitian $H$ , $Z = X H / 2$ gives $D F_{X} [Z] = H$ . Hence the level is regular, and [F2] gives the displayed tangent kernel.

2.1givenstep 1.1algebra

Set $S = Herm (X^{*} G)$ . Then $X^{*} (G - X S) + (G - X S)^{*} X = 0$ , so $G - X S$ is tangent. For tangent $Z$ , put $K = X^{*} Z$ , which is skew-Hermitian. For Hermitian $S$ , $\overline{tr (S K)} = - tr (S K)$ by adjoint and cyclic-trace identities; thus $Re tr (S K) = 0$ . Therefore $X S$ is normal in the real Frobenius metric, and the stated formula is the Stiefel orthogonal projection.

2.2F1F3givenstep 1.1algebra

A curve $Q (t)$ in $U (r)$ through $I_{r}$ has $Ω = Q^{'} (0)$ skew-Hermitian by differentiating $Q (t)^{*} Q (t) = I_{r}$ . Conversely, for any skew-Hermitian $Ω$ , the Cayley curve $Q (t) = (I_{r} - \frac{t}{2} Ω)^{- 1} (I_{r} + \frac{t}{2} Ω)$ is defined near zero, satisfies $Q (t)^{*} Q (t) = I_{r}$ , and has $Q (t) = I_{r} + t Ω + O (t^{2})$ ; hence the vertical tangent directions are exactly $X Ω$ . Every Stiefel tangent decomposes as $Z = X Ω + (I - X X^{*}) Z$ with $Ω = X^{*} Z$ . The second summand is horizontal. For any vertical $X Ω$ and horizontal $W$ , $⟨ X Ω, W ⟩_{R} = Re tr (Ω^{*} X^{*} W) = 0$ , so the summands are orthogonal. The decomposition is unique, and the graph-chart quotient identifies its horizontal component with the quotient tangent.

3.1F1givenstep 2.2algebra∎

The matrix $(I - X X^{*}) G$ is horizontal. For every horizontal $Z$ , $⟨ G - (I - X X^{*}) G, Z ⟩_{R} = Re tr ((X X^{*} G)^{*} Z) = Re tr (G^{*} X X^{*} Z) = 0$ since $X^{*} Z = 0$ . Thus it is the horizontal orthogonal projection. Under $(X, G) \mapsto (X Q, G Q)$ for unitary $Q$ , it transforms to $(I - X X^{*}) G Q$ , the same tangent vector in the changed frame.

Depends on

Used by

Polar retraction on the complex Grassmannian Definition
Complex finite-basis PC-VAR optimization on Grassmann Example
Gradient of a unitary-invariant Hermitian projector energy Proposition
The complex Grassmann polar map is a first-order retraction Theorem

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Peter A. Absil, Robert Mahony, and Rodolphe Sepulchre, Optimization Algorithms on Matrix Manifolds, Chapter 3 (2008) (standard reference, not scraped)
Nicolas Boumal, An Introduction to Optimization on Smooth Manifolds (2023 prepublication) (standard reference, not scraped)
Nicholas D. S. Manton, Optimization algorithms exploiting unitary constraints, IEEE Transactions on Signal Processing 50 (2002), §§IV and VI (standard reference, not scraped)