Real higher rank numerical ranges and ellipsoids

Abstract

We consider real versions of the classical numerical range and higher rank numerical ranges of a matrix. Our motivations come from descriptions of geometric structures in Euclidean space such as ellipsoids. We establish a number of basic results on these numerical ranges, comparing and delineating from their complex counterparts. We prove a two-dimensional elliptical range theorem for the joint real numerical range. We show that the joint numerical range and the real joint numerical range respectively of a set of m Hermitian matrices of order n is contained in an affine subspace of dimension n2−1 and 12(n2+n−2) respectively.