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

TL;DR

This paper establishes tight lower bounds for the trace norm of real symmetric and Hermitian matrices with zero diagonal, relating it to the entry-wise L1-norm, and provides bounds on the distance to diagonal matrices.

Contribution

It derives exact minimal ratios of trace norm to entry-wise L1-norm for zero-diagonal symmetric and Hermitian matrices, extending understanding of matrix norms and distances.

Findings

Minimum ratio for real symmetric matrices is 2/n.

Minimum ratio for Hermitian matrices is tan(π/2n).

Provides bounds on spectral distance to diagonal matrices.

Abstract

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.