When is the Adjoint of a Matrix a Low Degree Rational Function in the Matrix?

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