Plotting the Generalized Numerical Ranges Associated with the Compact Classical Groups

We prove an ellitpical range theorem for the numerical range associated with SO(3). The geometry behind the elliptical range theorem is explained. The result is analogous to the well-known elliptical range theorem of the C-numerical range associated with U(2) or SU(2). We then apply the result to give some necessary conditions of the sharp points of the C-numerical range assoicated with SO(n) when n 3, C = e 1 e T 2 ? e 2 e T 1 , and e 1 and e 2 are the rst two vectors of the standard basis of R n. Some properties of the generalized numerical ranges associated with the compact classical groups SO(n) and Sp(n) are obtained, namely, symmetry, projection, inclusion relation and direct sum. We provide MATLAB programs for plotting the numerical ranges associated with SO(n) and Sp(n). Various examples are discussed.