Minimum trace norm of real symmetric and Hermitian matrices with zero diagonal

We obtain tight lower bounds for the trace norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_1$ of some matrices with diagonal zero, in terms of the entry-wise $L^1$-norm (denoted by $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{(1)}$). 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 $\frac{\mathrm{\parallel }A{\mathrm{\parallel }}_1}{\mathrm{\parallel }A{\mathrm{\parallel }}_{(1)}}$ 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.