DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Compatible left and right eigenvectors for a simple eigenvalue

Definition

Let $A \in C^{n \times n}$ , equipped with the standard Hermitian inner product, and let $λ \in C$ be a simple eigenvalue of $A$ . A nonzero vector $x$ is a right eigenvector for $λ$ if

$A x = λ x .$

A nonzero vector $y$ is a left eigenvector for $λ$ if

$y^{*} A = λ y^{*},$

equivalently $A^{*} y = \overline{λ} y$ .

Such $x$ and $y$ are called compatible left and right eigenvectors for the simple eigenvalue $λ$ .

Depends on

Used by

For a simple eigenvalue, left and right eigenvectors pair nontrivially and may be normalized by y^*x=1 Proposition

Dependency tree · two levels

14 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

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