  We show that every self--adjoint matrix B of trace 0 can be realized as
B=T+T^* for a nilpotent matrix T of norm no greater than K times the norm of B,
for a constant K that is independent of matrix size. More particularly, if D is
a diagonal, self--adjoint n-by-n matrix of trace 0, then there is a unitary
matrix V=XU_n, where X is an n-by-n permutation matrix and U_n is the n-by-n
Fourier matrix, such that the upper triangular part, T, of the conjugate V^*DV
of D has norm no greater than K times the norm of D. This matrix T is a
strictly upper triangular Toeplitz matrix such that T+T^*=V^*DV. We apply this
and related results to give partial answers to questions about real parts of
quasinilpotent elements in finite von Neumann algebras.
