Numerical range of a normal compression

Abstract

Let N be an n-by-n diagonal matrix whose distinct eigenvalues form corners of their convex hull, let x be a vector in with nonzero components, and let A be the compression of N to the orthogonal complement of x . In this article, we study the properties of the eigenvalues and the numerical range of A and show that in many ways they are analogous to the ones for unitary N . The approach via the diagonal form of N yields a much simpler proof for some of the main results in this area.