Work

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

Year: 2025

Type: article

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 L 1 {L}^{1} -norms ( ‖ ⋅ ‖ ( 1 ) ) \left(\Vert \cdot {\Vert }_{\left(1)}) . For the space of nonzero real symmetric matrices of order n n , we prove that the minimum possible ratio ‖ A ‖ 1 ‖ A ‖ ( 1 ) \frac{\Vert A{\Vert }_{1}}{\Vert A{\Vert }_{\left(1)}} is exactly 2 n \frac{2}{n} . In the Hermitian case, this minimum ratio is given by tan π 2 n \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 n 2 \frac{n}{2} from a diagonal matrix, while the corresponding bound for Hermitian matrices is cot <

Source: Special Matrices

DOI: 10.1515/spma-2025-0040

Authors: Mostafa Einollahzadeh, Mohammad Ali Nematollahi

Institutions: Isfahan University of Technology, Institute for Research in Fundamental Sciences, Fasa University

Language: English

Topic: Matrix Theory and Algorithms

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $1,187

Funders: School of Mathematics, Institute for Research in Fundamental Sciences, Institute for Research in Fundamental Sciences