Componentwise pseudospectrum of a matrix
For \(\varepsilon > 0\), the \(\varepsilon\)-pseudospectrum of an \(n \times n\) matrix \(A\) is defined as \(\Lambda_\varepsilon(A) = \{ \lambda \in \mathbb{C} : \exists \Delta \in \mathbb C^{n \times n}\) such that \(\| \Delta \| _2 \leq \varepsilon\) and \(A + \Delta - \lambda I\) is singular\(\} = \{ \lambda \in \mathbb{C} : \sigma_{\min}(A - \lambda I) \leq \varepsilon \} = \{ \lambda \in \mathbb{C} : \| (A - \lambda I )^{-1} \geq \frac{1}{\varepsilon} \}\). The authors propose to define the componentwise \(\varepsilon\)-pseudospectrum by \(\Lambda_\varepsilon(A,E) = \{ \lambda \in \mathbb{C} : \exists \Delta \in \mathbb C^{n \times n}\) such that \(\| \Delta \| _2 \leq \varepsilon E\) and \(A + \Delta - \lambda I\) is singular\(\} = \{ \lambda \in \mathbb{C} : \sigma(A - \lambda I,E) \leq \varepsilon \}\). Furthermore, componentwise \(\varepsilon\)-pseudospectrum-equivalents for the third term in the definition of an \(\varepsilon\)-pseudospectrum are discussed and numerical examples are given.
scientific article; zbMATH DE number 2081759