TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-31

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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For a unit vector $x$ , the smallest perturbation making $(μ, x)$ an exact eigenpair has spectral norm $∥ A x - μ x ∥_{2}$

Statement

Let $F \in {R, C}$ , let $A \in M_{n} (F)$ , let $μ \in F$ , and let $x \in F^{n}$ satisfy $∥ x ∥_{2} = 1$ . Put $r : = A x - μ x$ . Then

$η (A, μ, x) = ∥ r ∥_{2} .$

In particular, the rank-one perturbation $E : = - r x^{*}$ attains the infimum.

Facts & Assumptions

Given: A unit vector $x \in F^{n}$ , a scalar $μ \in F$ , a matrix $A \in M_{n} (F)$ , and the residual $r = A x - μ x$ , where $F \in {R, C}$ .

[L1]

The residual and the normwise backward error are defined by $r = A x - μ x$ and $η (A, μ, x) = \inf {∥ E ∥_{2} : (A + E) x = μ x}$ (The residual $r = A x - μ x$ and the normwise backward error of an approximate eigenpair).

[L2]

The induced operator norm satisfies $∥ E y ∥_{2} \leq ∥ E ∥_{2} ∥ y ∥_{2}$ (The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||).

[L3]

Cauchy--Schwarz gives $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

Proof

technique · direct

1.1L1L2algebra

If $(A + E) x = μ x$ , then $E x = - (A x - μ x) = - r$ . Because $∥ x ∥_{2} = 1$ , [L2] gives $∥ E ∥_{2} \geq ∥ E x ∥_{2} = ∥ r ∥_{2} .$ So every admissible perturbation has norm at least $∥ r ∥_{2}$ .

1.2L1L3constructalgebra

Define $E : = - r x^{*}$ . Then $E x = - r (x^{*} x) = - r,$ so $(A + E) x = μ x$ . For any unit vector $y$ , [L3] gives $∥ E y ∥_{2} = ∥ r ∥_{2} ∣ x^{*} y ∣ \leq ∥ r ∥_{2},$ with equality at $y = x$ . Hence $∥ E ∥_{2} = ∥ r ∥_{2}$ .

2.1step 1.1step 1.2∎

Step 1.1 gives the lower bound and step 1.2 attains it, so $η (A, μ, x) = ∥ r ∥_{2}$ .

Depends on

Used by

Dependency tree · two levels

9 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

Netlib Templates, Numerical Stability and Conditioning (standard reference, not scraped)

For a unit vector xx, the smallest perturbation making (μ,x)(μ,x) an exact eigenpair has spectral norm ∥Ax−μx∥2∥Ax−μx∥2​