Note on a paper of thompson: the congruence numerical range

Given n×n Complex matrices A, Cdefine the C-congruence numerical range of A to be the set [ILM0001]. R.C. Thompson has characterized R_C(A) when [ILM0002] are fixed complex numbers. In this note. we obtain some analogous results about R_t(A) when C is skew symmmetric and a simple proof of the result of Thompson is given.Moreover, we characterize a certain set of partial off diagonals under congruence unitary transformation.