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Result · E-REPEATED-EIGEN-DEGENERACY

Repeated eigenvalues weaken E geometry

Proved herepublication: internal

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Meaning

A repeated minimum eigenvalue makes one-transfer first-order geometry non-identifying.

Statement

For a one-transfer rank-two direction and minimum eigenspace dimension r>=2, the directional derivative is <=0; one-point first-order stability can become automatic at the degeneracy.

Relationships

Rests on

The full chain beneath it

Enables

Raises

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Local graph
Local map of Repeated minimum eigenvalues make one-transfer first-order E geometry non-identifying2 prerequisites to the left, 1 results it enables or questions it raises to the right, 0 boundary cases below.Infinitesimal cell-transfer identityInfinitesimal cell-tr…E-optimality supergradient structureE-optimality supergra…Population E common-supergradient geometryPopulation E common-s…Repeated minimum eigenvalues make one-transfer first-order E geometry non-identifyingRepeated minimum eige…
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

Proof

E2. Repeated-eigenvalue one-transfer degeneracy

For one infinitesimal transfer ΔI=aa⊤−bb⊤\Delta I=aa^\top-bb^\top,

dλmin⁡(I;ΔI)=λmin⁡(V⊤ΔIV)≤0d\lambda_{\min}(I;\Delta I) =\lambda_{\min}(V^\top\Delta I V)\le0

whenever the minimum eigenspace dimension is r≥2r\ge2. Thus one-point first-order stability can become automatic/non-identifying exactly where E tends to equalize weak directions.

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