A comparison of eigenvalue condition numbers for matrix polynomials
For researchers in numerical linear algebra, this work clarifies the relationship between existing condition numbers, but the contribution is incremental.
This paper compares different eigenvalue condition numbers for matrix polynomials, showing that two homogeneous condition numbers are essentially the same for moderate-degree polynomials and extend the Wilkinson-like condition number to zero and infinite eigenvalues.
In this paper, we consider the different eigenvalue condition numbers for matrix polynomials used in the literature and we compare them. One of these condition numbers is a generalization of the Wilkinson condition number for the standard eigenvalue problem. This number has the disadvantage of only being defined for finite eigenvalues. In order to give a unified approach to all the eigenvalues of a matrix polynomial, both finite and infinite, two (homogeneous) condition numbers have been defined in the literature. In their definition, very different approaches are used. One of the main goals of this note is to show that, when the matrix polynomial has a moderate degree, both homogeneous numbers are essentially the same and one of them provides a geometric interpretation of the other. We also show how the homogeneous condition numbers compare with the "Wilkinson-like" eigenvalue condition number and how they extend this condition number to zero and infinite eigenvalues.