Alphabeta Math
CorollaryStatement: Literature-sourcedProof: Literature-sourcedaudited 2026-09-04
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Index and coindex swap under negation

Statement

Let f:M→R be smooth and let p be a critical point of f. Then p is also a critical point of −f, the nullity is unchanged, and the index and coindex are exchanged:

null⁡p(−f)=null⁡p(f),ind⁡p(−f)=coind⁡p(f),coind⁡p(−f)=ind⁡p(f).

Facts & Assumptions

Given: A smooth function f:M→R and a critical point p of f.

[F1]

Nullity, index, and coindex are defined from the Hessian by kernel, negative-definite subspaces, and positive-definite subspaces (Nondegenerate critical points, nullity, index, and coindex).

Proof

technique · direct sign reversal
1.1

Because differentiation is linear, Hess⁡p(−f)=−Hess⁡p(f).

F1givenalgebra
2.1

Multiplication by −1 does not change the kernel of a bilinear form, so the nullity is unchanged.

F1step 1.1
2.2

A subspace is negative definite for −Hess⁡p(f) exactly when it is positive definite for Hess⁡p(f), and similarly with "positive" and "negative" exchanged. Therefore the index and coindex swap by [F1].

F1step 1.1
3.1

Hence negating f preserves nullity and exchanges index with coindex.

step 2.1step 2.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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