Abstract We establish tight lower bounds for the trace norm (‖ ⋅ ‖ 1) (₁) of real symmetric and Hermitian matrices with zero diagonal entries in terms of their entrywise L 1 L^1 -norms (‖ ⋅ ‖ (1) ) ( (₁) ). For the space of nonzero real symmetric matrices of order n n, we prove that the minimum possible ratio ‖ A ‖ 1 ‖ A ‖ (1) A ₁ A{ (₁) } is exactly 2 n 2n. In the Hermitian case, this minimum ratio is given by tan π 2 n (-0. 75em{0ex}, 2n). 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 n 2 n2 from a diagonal matrix, while the corresponding bound for Hermitian matrices is cot π 2 n (-0. 75em{0ex}, 2n). Applications to graph energy and quantum coherence are discussed, highlighting implications for algebraic graph theory and quantum resource theory.
Einollahzadeh et al. (Wed,) studied this question.