Lesson plan
Pseudospectrum
In mathematics, the pseudospectrum of an operator is a set containing the spectrum of the operator and the numbers that are "almost" eigenvalues. Knowledge of the pseudospectrum can be particularly useful for understanding non-normal operators and their eigenfunctions. The ε-pseudospectrum of a matrix A consists of all eigenvalues of matrices which are ε-close to A: : \Lambda_\epsilon(A) = \{\lambda \in \mathbb{C} \mid \exists x \in \mathbb{C}^n \setminus \{0\}, \exists E \in \mathbb{C}^{n \times n} \colon (A+E)x = \lambda x, \|E\| \leq \epsilon \}. Numerical algorithms which calculate the eigenvalues of a matrix give only approximate results due to rounding and other errors. These errors can be described with the matrix E. More generally, for Banach spaces X,Y and operators A: X \to Y , one can define the \epsilon -pseudospectrum of A (typically denoted by \text{sp}_{\epsilon}(A) ) in the following way : \text{sp}_{\epsilon}(A) = \{\lambda \in \mathbb{C} \mid \|(A-\lambda I)^{-1}\| \geq 1/\epsilon \}. where we use the convention that \|(A-\lambda I)^{-1}\| = \infty if A - \lambda I is not invertible.
Article summary
In mathematics, the pseudospectrum of an operator is a set containing the spectrum of the operator and the numbers that are "almost" eigenvalues. Knowledge of the pseudospectrum can be particularly useful for understanding non-normal operators and their eigenfunctions. The ε-pseudospectrum of a matrix A consists of all eigenvalues of matrices which are ε-close to A: : \Lambda_\epsilon(A) = \{\lambda \in \mathbb{C} \mid \exists x \in \mathbb{C}^n \setminus \{0\}, \exists E \in \mathbb{C}^{n \times n} \colon (A+E)x = \lambda x, \|E\| \leq \epsilon \}. Numerical algorithms which calculate the eigenvalues of a matrix give only approximate results due to rounding and other errors. These errors can be described with the matrix E. More generally, for Banach spaces X,Y and operators A: X \to Y , one can define the \epsilon -pseudospectrum of A (typically denoted by \text{sp}_{\epsilon}(A) ) in the following way : \text{sp}_{\epsilon}(A) = \{\lambda \in \mathbb{C} \mid \|(A-\lambda I)^{-1}\| \geq 1/\epsilon \}. where we use the convention that \|(A-\lambda I)^{-1}\| = \infty if A - \lambda I is not invertible.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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