We study a minimum problem and associated maximum problem for finite,
complex, self-adjoint Toeplitz matrices. If $A$ is such a matrix, of size
$(N+1)$-by-$(N+1)$, we identify $A$ with the operator it represents on $P_N$,
the space of complex polynomials of degrees at most $N$, with the usual Hilbert
space structure it inherits as a subspace of $L^2$ of the unit circle. The
operator $A$ is the compression to $P_N$ of the multiplication operator on
$L^2$ induced by any function in $L^{\infty}$ whose Fourier coefficients of
indices between $-N$ and $N$ match the matrix entries of $A$.