Interval Analysis

7Eigenvalue problems and related ones

In the present chapter we deal with the verification and enclosure of eigenpairs (x^∗, λ^∗) – primarily for a single real matrix A ∈ ℝ^n×n but on second glance also for an interval matrix [A] ∈ 𝕀ℝ^n×n. In the latter case ‘verification and enclosure’ means that we look for an interval vector [x] and an interval [λ] such that for each matrix eigenvalue problem for à ∈ [A] there is an eigenpair . We will denote this as ‘interval eigenvalue problem for [A]’.

Since eigenvectors x^∗ are not unique, we will normalize them in many cases by fixing the n-th component to be equal to one, i.e.,

In this case we will speak ...