Generalizations of the field of values useful in the study of polynomial functions of a matrix
When estimating \(\|f(A) \|\) for various functions \(f\), where \(A\) is an \(n \times n\) matrix, it is useful to associate \(A\) with some set in the complex plane and relate \(\|f(A) \|\) to the size of \(f\) on this set. For normal matrices an appropriate set is the spectrum of \(A\). For nonnormal matrices the situation is less clear. In this paper polynomials of a fixed degree \(k\) or less satisfying \[ \|p(A) \|\geq \max_{z \in \Omega} |p(z) | \] are considered and sets \(\Omega\) are looked for. The largest such set \(\Omega\) was discussed by \textit{O. Nevanlinna} [Convergence of iterations for linear equations (Birkhäuser, Basel) (1993; Zbl 0846.47008)] and referred as the polynomial numerical hull of degree \(k\). These sets are for nonnormal matrices, in some way, the analog of eigenvalues for a normal matrix. For \(k=1\), each of these sets is the field of values \(A\) and for \(k \geq m\), where \(m\) is the degree of the minimal polynomial of \(A\), they are shown to be the spectrum of \(A\). For \(1 < k < m\) the sets are intermediate between the field of values and the spectrum. Several equivalent characterizations of these sets are presented and for some examples (Jordan block, Gauss-Seidel matrix, the Ehrenfest's urn and Grcar matrix) the calculation of the polynomial numerical hull is carried out. One possible application of the analysis of polynomial functions of a matrix is in describing the convergence rate of the GMRES algorithm for solving linear systems \(Ax=b\).
scientific article; zbMATH DE number 1790235