ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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The directional derivative of a simple singular value is the real part of $u^{*} H v$

Example

Take

$A = (\begin{array}{cc} 2 & 0 \\ 0 & 1 \end{array}), H = (\begin{array}{cc} 1 & 1 \\ 0 & 0 \end{array}) .$

The largest singular value is $σ = 2$ , with unit left and right singular vectors $u = v = e_{1}$ . Therefore

$D σ (A) [H] = Re (u^{*} H v) = 1.$

Facts & Assumptions

Given: The matrices $A$ and $H$ above.

[L1]

For a simple positive singular value, $D σ (A) [H] = Re (u^{*} H v)$ (If $σ > 0$ is a simple singular value with left and right singular vectors $u, v$ , then its real directional derivative is $Re (u^{*} H v)$ ).

Verification

technique · direct

1.1

Because $A$ is diagonal with entries $2$ and $1$ , its singular values are $2$ and $1$ . The largest one is simple, with $u = v = e_{1}$ .

algebra

2.1

Applying [L1] gives $D σ (A) [H] = Re (e_{1}^{*} H e_{1}) = 1$ .

L1step 1.1algebra∎

Depends on

If $\sigma>0$ is a simple singular value with left and right singular vectors $u,v$, then its real directional derivative is $\operatorname{Re}(u^*Hv)$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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 directional derivative of a simple singular value is the real part of u∗Hvu∗Hv