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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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At a symmetric crossing, an ordered eigenvector branch cannot remain differentiable

Statement refuted

Across a symmetric eigenvalue crossing, one can keep the ordered eigenvector branch differentiable by a clever normalization.

The family

$A (t) = (\begin{array}{cc} t & 0 \\ 0 & - t \end{array})$

shows that no normalization can fix the fact that the top eigendirection swaps from $span {e_{2}}$ to $span {e_{1}}$ at $t = 0$ .

Facts & Assumptions

Given: The symmetric crossing family $A (t) = (\begin{array}{cc} t & 0 \\ 0 & - t \end{array})$ .

[L1]

An ordered eigenvector branch need not be differentiable through a crossing (An ordered eigenvector branch need not extend differentiably through an eigenvalue crossing).

Counterexample

technique · direct

1.1

For $t < 0$ , the larger eigenvalue is $- t$ with eigendirection $span {e_{2}}$ , while for $t > 0$ the larger eigenvalue is $t$ with eigendirection $span {e_{1}}$ .

algebra

2.1

Any normalization still has to represent those two different eigendirections on the two sides of the crossing, so the ordered branch cannot be continuous or differentiable through $t = 0$ . This is exactly the phenomenon recorded in [L1].

L1step 1.1∎

Depends on

An ordered eigenvector branch need not extend differentiably through an eigenvalue crossing

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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)