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

Abstract

We establish tight lower bounds for the trace norm (‖⋅‖1)\left(\Vert \cdot {\Vert }_{1}) of real symmetric and Hermitian matrices with zero diagonal entries in terms of their entrywise L1{L}^{1}-norms (‖⋅‖(1))\left(\Vert \cdot {\Vert }_{\left(1)}). For the space of nonzero real symmetric matrices of order nn, we prove that the minimum possible ratio ‖A‖1‖A‖(1)\frac{\Vert A{\Vert }_{1}}{\Vert A{\Vert }_{\left(1)}} is exactly 2n\frac{2}{n}. In the Hermitian case, this minimum ratio is given by tanπ2n\tan \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{\pi }{2n}\right). Through duality arguments, we derive sharp upper bounds for the spectral norm distance between a matrix and the space of diagonal matrices. For instance, any real symmetric matrix with off-diagonal entries bounded by 1 in absolute value lies within a spectral norm distance of n2\frac{n}{2} from a diagonal matrix, while the corresponding bound for Hermitian matrices is cotπ2n\cot \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{\pi }{2n}\right). Applications to graph energy and quantum coherence are discussed, highlighting implications for algebraic graph theory and quantum resource theory.

Keywords

• trace norm
• matrix norm inequality
• graph energy
• nearest diagonal matrix
• quantum coherence
• 15a42
• 05c50