DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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.

The real Frechet derivative on real and complex matrix spaces with the Frobenius norm

Definition

Let $U$ be an open subset of a real matrix space $R^{m \times n}$ or of a complex matrix space $C^{m \times n}$ viewed as a real vector space, and let $F : U \to W$ take values in a finite-dimensional real normed vector space $(W, ∥ \cdot ∥_{W})$ . We say that $F$ is real Fr'echet differentiable at $A \in U$ if there is a real-linear map $L$ such that

$\frac{∥ F (A + H) - F (A) - L [H] ∥_{W}}{∥ H ∥_{F}} ⟶ 0 (H \to 0, H \neq 0) .$

That map $L$ is the real Fr'echet derivative of $F$ at $A$ and is denoted $D F (A)$ .

Depends on

The Frobenius norm $\lVert A\rVert_F=(\sum_{i,j}|a_{ij}|^2)^{1/2}$ on real or complex matrices

Used by

Dependency tree · two levels

6 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

Alan Edelman and Steven G. Johnson, Matrix Calculus for Machine Learning and Beyond (standard reference, not scraped)