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

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The simple spectral projector is unchanged by nonzero rescalings of the left and right eigenvectors

Statement

Let $P = x y^{*} / (y^{*} x)$ be the simple spectral projector of a simple eigenvalue. If $α, β \neq 0$ , then the projector formed from $\tilde{x} = α x$ and $\tilde{y} = β y$ is still $P$ .

Facts & Assumptions

Given: A simple spectral projector $P = x y^{*} / (y^{*} x)$ and nonzero scalars $α, β$ .

[F1]

The projector attached to compatible eigenvectors is $x y^{*} / (y^{*} x)$ (The simple spectral projector $P = x y^{*} / (y^{*} x)$ ).

Proof

technique · direct

1.1

By [F1], one has $y^{*} x \neq 0$ . Since also $α, β \neq 0$ , the rescaled denominator satisfies ${\tilde{y}}^{*} \tilde{x} = \overline{β} α y^{*} x \neq 0.$ Therefore the rescaled projector is well defined, and using [F1] again gives $\frac{\tilde{x} {\tilde{y}}^{*}}{{\tilde{y}}^{*} \tilde{x}} = \frac{(α x) (\overline{β} y^{*})}{\overline{β} α y^{*} x} = \frac{x y^{*}}{y^{*} x} = P .$

F1algebra

2.1

Thus nonzero rescaling changes numerator and denominator by the same nonzero factor and leaves the spectral projector unchanged.

step 1.1∎

Depends on

The simple spectral projector $P=xy^*/(y^*x)$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

David Bindel, CS 6210: Matrix Computations - Perturbation theory (standard reference, not scraped)