Exclusion and Inclusion Regions for the Eigenvalues of a Normal Matrix

Abstract

Assume $N\in {\mathbb C}^{n \times n}$ is a square matrix with the characteristic polynomial p(z)=f(x,y)+ig(x,y). Viewing the spectrum $\sigma(N)$ of $N$ as an algebraic subvariety of ${\mathbb R}^2$, by Bezout's theorem, the degrees of f and g seem to be unnecessarily high for locating $\sigma(N)$. Starting from this observation, we employ real analytic techniques to find the spectrum of a normal matrix N. At most 1-dimensional information is obtained with polyanalytic polynomials of degree not exceeding $\sqrt{2n}$. This is achieved by performing only matrix-vector products with an algorithm relying on a recurrence with a very slowly growing length. For large problems three practical alternatives are proposed via computing Ritz values, eigenvalue exclusion, and eigenvalue inclusion regions.