Modifications of the Oettli-Prager Theorem with Application to the Eigenvalue Problem

Abstract

In this paper we consider the eigen pair set

$${E_\mathcal{P}}: = {\text{\{ (x,}}\lambda {\text{)|Ax = }}\lambda {\text{x,}} {\text{x}} \ne {\text{0,}} {\text{A}} \in {\text{[A],}} {\text{A}} {\text{with}} {\text{property}} \mathcal{P}{\text{\} ,}}$$

(1)

where [A] is a given real n x n interval matrix (cf. Alefeld and Herzberger (1983), e.g., for interval analysis) and $\mathcal{P}$ is some fixed property such as symmetry, Toeplitz form, etc.. Before we study this set in greater detail we mention other ones which are related to it: When dealing with systems of linear equations

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A} x = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{b} , \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A} \in {\mathbb{R}^{n \times n}}, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{b} \in {\mathbb{R}^n}$$

(2)

(ℝ^{n x n} set of real n x n matrices, ℝⁿ set of real vectors with n components) there sometimes occurs the problem of varying the input data $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{b}$ within certain tolerances and looking for the set S of the resulting solutions x*.