Exclusion and inclusion regions for the eigenvalues of a normal matrix (Q2784398)

Let \(A\) be an \(n\times n\) normal matrix with characteristic polynomial \(p(z)= f(x,y)+ ig(x,y)\). It is shown that because of the normality of \(A\), real analytic techniques yield bivariate polynomials such that generically, NEWLINE\[NEWLINE\deg f\deg g\leq 4n\tag{1}NEWLINE\]NEWLINE holds pairwise, so that in the light of Bézout's theorem, (1) is of correct order. The corresponding algorithm involves a recurrence with very slowly growing length.NEWLINENEWLINENEWLINEAlternatives for large problems proposed are Ritz calculations and methods for eigenvalue exclusion and inclusion regions. In practice, polyanalytic polynomials needed in the present context are generated by an Arnoldi type iteration.