PropositionStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Gradient of a unitary-invariant Hermitian projector energy

Statement

Let $1 \leq r \leq n$ , and let $E$ be real-valued and continuously differentiable on an open set of Hermitian $n \times n$ matrices containing the rank- $r$ orthogonal projectors. Suppose its real Frobenius gradient $K (P) = K (P)^{*}$ satisfies $D E (P) [H] = Re tr (K (P)^{*} H)$ for every Hermitian direction $H$ . Then $f ([X]) = E (X X^{*})$ is well-defined on ${Gr}_{C} (r, n)$ and has horizontal gradient $grad f ([X]) = 2 (I - X X^{*}) K (X X^{*}) X .$

Facts & Assumptions

Given: The real differentiable Hermitian-matrix functional and gradient in the Statement, and $X \in {St}_{C} (n, r)$ .

[F1]

The frame quotient is by the right unitary action, and complex matrix spaces use the real Frobenius pairing (Complex Stiefel and Grassmann optimization conventions).

[F2]

The horizontal tangent projection is $G \mapsto (I - X X^{*}) G$ (Complex Stiefel and Grassmann tangent projections).

[F3]

A Riemannian gradient is the metric dual of the differential (The Riemannian gradient is the metric dual of the differential).

[F4]

Complex Grassmann has local graph charts and frame representatives related by right unitary maps (Stiefel spaces, Grassmannians, and tautological bundles).

[F5]

The complex polar retraction has frame differential $D R_{X} (0) [Z] = Z$ for horizontal $Z$ (The complex Grassmann polar map is a first-order retraction).

Proof

technique · differentiate the projector map on horizontal directions, then project the ambient metric dual

1.1F1F4F5givenalgebra

For unitary $Q$ , $(X Q) (X Q)^{*} = X X^{*}$ , so $E (X X^{*})$ is independent of the representative. In the local graph charts [F4], it therefore defines a differentiable function $f$ on the quotient. Let $Y (t) = R_{X}^{p o l} (t Z)$ and $P (t) = Y (t) Y (t)^{*}$ . By [F5], $Y (0) = X$ and $Y^{'} (0) = Z$ , so differentiating gives $P (0) = P$ and $P^{'} (0) = Z X^{*} + X Z^{*}$ . Since $E$ is defined on an open neighborhood of $P$ , the chain rule gives the differential of $f$ along this Grassmann curve.

2.1F1F2F3givenstep 1.1algebra∎

Using $K = K^{*}$ , cyclicity of trace, and [F1], $d f_{[X]} [Z] = Re tr (K (Z X^{*} + X Z^{*})) = 2 Re tr ((K X)^{*} Z) = ⟨ 2 K X, Z ⟩_{R}$ . The horizontal metric dual is the projection of $2 K X$ . By [F2] it equals $2 (I - X X^{*}) K X$ , which is the claimed gradient by [F3].

Depends on

Used by

Complex finite-basis PC-VAR optimization on Grassmann Example

Dependency tree · two levels

17 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

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), §§VI–VII (standard reference, not scraped)