When is the Resolvent Like a Rank One Matrix?

For a square matrix [Formula: see text], the resolvent of [Formula: see text] at a point [Formula: see text] that is not an eigenvalue of [Formula: see text] is defined as [Formula: see text]. We consider the set of points [Formula: see text] where the relative difference in 2-norm between the resolvent and the nearest rank one matrix is less than a given number [Formula: see text]. We establish a relationship between this set and the [Formula: see text]-pseudospectrum of [Formula: see text], and we derive specific results about this set for Jordan blocks and for large banded Toeplitz matrices. We also derive disks (and other regions) about the eigenvalues of [Formula: see text] that are contained in this set, and this leads to some new results on regions about the eigenvalues that are contained in the [Formula: see text]-pseudospectrum of [Formula: see text]. In addition, we consider the set of points [Formula: see text] where the absolute value of the inner product of the left and right singular vectors corresponding to the largest singular value of the resolvent is less than [Formula: see text]. We demonstrate numerically that this set can be almost as large as the one where the relative difference between the resolvent and the nearest rank one matrix is less than [Formula: see text] and we give a partial explanation for this. Some possible applications are discussed.