TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-31

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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.
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The Frobenius least-squares objective has gradient $A^{} (A x - b)$ and Hessian $A^{} A$ in the vector variable

Statement

For fixed $A$ and $b$ , let

$f (x) : = \frac{1}{2} ∥ A x - b ∥_{2}^{2} .$

Then

$D f (x) [h] = Re ⟨ A^{*} (A x - b), h ⟩,$

so the gradient is $\nabla f (x) = A^{*} (A x - b)$ and the Hessian is the constant map $h \mapsto A^{*} A h$ .

Facts & Assumptions

Given: A fixed matrix $A$ , a fixed vector $b$ , a vector $x$ , and a direction $h$ .

[L1]

The quadratic form $x \mapsto \frac{1}{2} ∥ C x ∥_{2}^{2}$ has gradient $C^{*} C x$ and Hessian $C^{*} C$ (Matrix quadratic forms have the expected first derivative and Hessian).

Proof

technique · direct

1.1L1algebra

Write $f (x) = \frac{1}{2} ∥ A (x - A^{+} b) ∥_{2}^{2} + constant$ only heuristically; directly, $f (x + h) - f (x) = \frac{1}{2} ∥ A x - b + A h ∥_{2}^{2} - \frac{1}{2} ∥ A x - b ∥_{2}^{2} .$ Expanding yields $D f (x) [h] = Re ⟨ A x - b, A h ⟩ = Re ⟨ A^{*} (A x - b), h ⟩ .$

2.1step 1.1algebra∎

The gradient map from step 1.1 is $x \mapsto A^{*} (A x - b)$ , whose derivative is the constant linear map $h \mapsto A^{*} A h$ . Thus the Hessian is $A^{*} A$ .

Depends on

Used by

A small least-squares problem makes the gradient and Hessian formulas explicit Example

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

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

The Frobenius least-squares objective has gradient A∗(Ax−b)A∗(Ax−b) and Hessian A∗AA∗A in the vector variable