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Pseudospectra
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
The ε-pseudospectrum is a subset of the complex plane that always includes the spectrum, but can potentially contain points far from any eigenvalue. Unlike the spectrum, pseudospectra vary with choice of norm and, thus, for a given application one must take care to work in a physically appropriate norm. Unless otherwise noted, throughout this chapter we assume that A∈ℂn×n is a square matrix with complex entries, and that ║ · ║ denotes a vector space norm and the matrix norm it induces. When speaking of a norm associated with an inner product, we presume that adjoints and normal and unitary matrices are defined with respect to that inner product. All computational examples given here use the 2-norm.
Convergence analysis of iterative learning control using pseudospectra
Published in International Journal of Control, 2022
Zahra Shahriari, Bo Bernhardsson, Olof Troeng
In Bristow and Singler (2011), transient behaviour of ILC is analysed using pseudospectra. Pseudospectra is a mathematical tool which, as its name suggests, is complementary to the spectra of matrices or operators. Analysing the spectrum (or the set of eigenvalues) can be misleading in many applications. A number of examples are provided in Pseudospectra gateway (n.d.). The idea of pseudospectra can be motivated as follows. Let and denote the 2-norm. A point in the complex plane z is said to be an eigenvalue of if is non-invertible. The matrix is the resolvent of the matrix at z. For some practical purposes, it is not enough to distinguish between the points where the norm of the resolvent is finite or infinite. Rather, one could investigate how large the norm of the resolvent is at a certain point z. This leads to the first definition of pseudospectra (see Definition 2.5 in Section 2.2). Another definition of the pseudospectrum is a result of eigenvalue perturbation theory (Kato, 2013). If is a normal matrix, i.e. if , its eigenvalues are robust against perturbations. That is, for any perturbation matrix with the eigenvalues of the perturbed matrix are within an ε-ball neighborhood of the original eigenvalues of . For non-normal matrices, however, the eigenvalues can be highly sensitive to perturbation. In such cases, eigenvalue analysis could be misleading and the pseudospectra provide a better insight into the behaviour of the matrix. This leads to another, equivalent, definition of the pseudospectra: where Λ denotes the spectrum. In Reichel and Trefethen (1992) it is remarked that ‘A pseudo-eigenvalue, in other words, need not be near to any exact eigenvalue, but it is an exact eigenvalue of some nearby matrix’.