DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-31

How statement and proof provenance work

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The normwise condition number of a simple eigenvalue

Definition

Let $λ$ be a simple eigenvalue of $A$ . The normwise first-order condition number of $λ$ is

$κ (λ; A) : = \sup_{∥ H ∥_{2} = 1} ∣ D λ (A) [H] ∣,$

where, for any compatible nonzero right and left eigenvectors $x, y$ , we set

$D λ (A) [H] : = \frac{y^{*} H x}{y^{*} x} .$

The denominator is nonzero by For a simple eigenvalue, left and right eigenvectors pair nontrivially and may be normalized by $y^{*} x = 1$ . Any other compatible pair differs by nonzero rescalings because the two eigenspaces are one-dimensional, and those factors cancel from the quotient, so this definition is independent of the chosen pair. The later eigenvalue-derivative theorem shows that this functional is the actual derivative of every local simple eigenvalue branch.

Depends on

Used by

The normwise condition number of a simple eigenvalue is ‖x‖₂‖y‖₂/|y^*x| Proposition

Dependency tree · two levels

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Sources

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