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We obtain tight lower bounds for the trace norm $\Vert \cdot \Vert_1$ of somematrices 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 realsymmetric matrices $A$ of order $n$ with diagonal zero, the minimum value ofthe 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 Hermitianmatrices, is also obtained to be equal to $\tan(\frac{\pi}{2n})$. Theequivalent "dual" form of these results, give some upper bounds for thedistance to the nearest diagonal matrix for a given symmetric or Hermitianmatrix, when the distance is computed in the spectral norm.

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