PropositionStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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Trace and Frobenius-linear matrix functionals differentiate by inspection

Statement

For square matrices,

$D (tr) (A) [H] = tr (H) .$

For a fixed matrix $B$ , the real-valued Frobenius-linear functional

$Φ_{B} (A) : = Re tr (B^{*} A)$

satisfies

$D Φ_{B} (A) [H] = Re tr (B^{*} H) .$

Thus the Frobenius gradient of $Φ_{B}$ is $B$ .

Facts & Assumptions

Given: A matrix $A$ , a perturbation direction $H$ , and a fixed matrix $B$ .

[F1]

Real Fr'echet differentiability identifies the first-order linear term in $F (A + H) - F (A)$ (The real Frechet derivative on real and complex matrix spaces with the Frobenius norm).

Proof

technique · direct

1.1F1algebra

The trace is linear, so $tr (A + H) - tr (A) = tr (H)$ . Likewise, $Φ_{B} (A + H) - Φ_{B} (A) = Re tr (B^{*} H) .$ Each increment is already linear in $H$ .

2.1F1step 1.1∎

Therefore [F1] gives the displayed derivatives. The Frobenius gradient is the unique matrix $G$ satisfying $D Φ_{B} (A) [H] = Re tr (G^{*} H)$ for every $H$ , and step 1.1 shows that $G = B$ .

Depends on

Used by

Matrix quadratic forms have the expected first derivative and Hessian Proposition

Dependency tree · two levels

10 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)