TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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The derivative of the simple spectral projector is expressed by the reduced resolvent and the perturbation

Statement

Let $A (t)$ be differentiable, let $λ (t)$ be a simple eigenvalue branch, let $P (t)$ be the corresponding simple spectral projector, and let $S$ be the reduced resolvent at $t = 0$ . Then

$P^{'} (0) = - S A^{'} (0) P (0) - P (0) A^{'} (0) S .$

Facts & Assumptions

Given: A differentiable simple spectral projector branch $P (t)$ for a simple eigenvalue branch of $A (t)$ , and the reduced resolvent $S$ at $t = 0$ .

[L1]

The reduced resolvent satisfies $S P = P S = 0$ and $S (A - λ I) = (A - λ I) S = I - P$ (The reduced resolvent satisfies the standard projector and inverse identities on the complementary invariant subspace).

Proof

technique · direct

1.1

Differentiate $(A - λ I) P = 0$ at $t = 0$ : $(A^{'} (0) - λ^{'} (0) I) P + (A - λ I) P^{'} = 0.$ Left-multiplying by $S$ and using [L1] together with $S P = 0$ gives $(I - P) P^{'} = - S A^{'} (0) P .$ Differentiating $P (A - λ I) = 0$ and right-multiplying by $S$ similarly gives $P^{'} (I - P) = - P A^{'} (0) S .$

L1givenalgebra

2.1

Differentiating $P^{2} = P$ gives $P^{'} = P^{'} P + P P^{'}$ . Because $P P^{'} P = 0$ , this decomposes as $P^{'} = (I - P) P^{'} P + P P^{'} (I - P)$ . Substituting the two identities from step 1.1 yields $P^{'} = - S A^{'} (0) P - P A^{'} (0) S,$ which is the claimed formula.

L1step 1.1algebra∎

Depends on

Used by

Different gauge choices change the eigenvector derivative but not the eigenvalue derivative or projector derivative Example

Dependency tree · two levels

7 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

Benjamin Texier, Basic matrix perturbation theory (standard reference, not scraped)