CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-31

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

Statement refuted

An eigenvector branch chosen by ordering eigenvalues can always be continued differentiably through a crossing.

The family

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

shows otherwise: the eigendirection belonging to the larger eigenvalue is $span {e_{1}}$ for $t > 0$ and $span {e_{2}}$ for $t < 0$ .

Facts & Assumptions

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

[L1]

Simple eigenpair branches exist locally only while the eigenvalue stays simple (A simple eigenvalue and a gauge-fixed right eigenvector admit local $C^{1}$ branches in the underlying real matrix space).

Counterexample

technique · direct

1.1

For $t > 0$ , the larger eigenvalue is $t$ with eigendirection $span {e_{1}}$ . For $t < 0$ , the larger eigenvalue is $- t$ with eigendirection $span {e_{2}}$ . At $t = 0$ , the eigenvalue $0$ has multiplicity $2$ .

algebra

2.1

Any branch chosen by ordering the eigenvalues by size must jump from $span {e_{2}}$ on the left to $span {e_{1}}$ on the right, so it is not even continuous, let alone differentiable, through the crossing. This agrees with [L1], which applies only on the simple-spectrum locus.

L1step 1.1∎

Depends on

A simple eigenvalue and a gauge-fixed right eigenvector admit local $C^1$ branches in the underlying real matrix space

Used by

At a symmetric crossing, an ordered eigenvector branch cannot remain differentiable Counterexample

Dependency tree · two levels

4 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)