Simple eigenvalue

An eigenvalue is simple when it has algebraic multiplicity one.

Table of contents
- Eigenvalue sensitivity Simple eigenvalue
  - First-order perturbation of a simple eigenvalue Eigenvalue sensitivity

Eigenvalue sensitivity ( $s (λ)$ )

For unit left eigenvector

u

and right eigenvector

v

of a real matrix at a simple eigenvalue

λ

, its normwise sensitivity is

s (λ) = \frac{1}{∣ u ^{T} v ∣} .

It measures the first-order amplification of a matrix perturbation into an eigenvalue perturbation.

First-order perturbation of a simple eigenvalue

If

A (t) = A + tE

and

λ (t)

is a differentiable function defining a simple eigenvalue, then

λ^{'} (0) = \frac{u ^{T} E v}{u ^{T} v} .

The Cauchy-Schwarz inequality and the operator norm give

∣ λ^{'} (0) ∣ \leq ∥ E ∥_{2} s (λ) .

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