When is the adjoint of a matrix a low degree rational function in the matrix?

Abstract

We show that the adjoint of a matrix with respect to a given inner product is a rational function in the matrix, if and only if the matrix is normal with respect to the inner product. We consider such matrices and analyze the McMillan degrees of the rational functions such that the matrix adjoint is a rational function in the matrix. We introduce the McMillan degree of the matrix as the smallest among these degrees, characterize this degree in terms of the number and distribution of the matrix eigenvalues, and compare the McMillan degree with the normal degree of the matrix, which is defined as the smallest degree of a polynomial for which the matrix adjoint is a polynomial in the matrix. We show that unless the matrix eigenvalues lie on a single circle in the complex plane, the ratio of McMillan degree and normal degree of the matrix is bounded by a small constant that depends neither on the number nor on the distribution of the matrix eigenvalues. Our analysis is motivated by applications in the area of short recurrence Krylov subspace methods.