  We obtain tight lower bounds for the trace norm $\Vert \cdot \Vert_1$ of some
matrices with diagonal zero, in terms of the entry-wise $L^1$-norm (denoted by
$\Vert \cdot \Vert_{(1)}$). It is shown that on the space of nonzero real
symmetric matrices $A$ of order $n$ with diagonal zero, the minimum value of
the quantity $\frac{\Vert A\Vert_1}{\Vert A\Vert_{(1)}}$ is equal to
$\frac{2}{n}$. The answer of the similar problem in the space of Hermitian
matrices, is also obtained to be equal to $\tan(\frac{\pi}{2n})$. The
equivalent "dual" form of these results, give some upper bounds for the
distance to the nearest diagonal matrix for a given symmetric or Hermitian
matrix, when the distance is computed in the spectral norm.
