2016/05/27 by Vladimir Nikiforov, Nikiforov, V.
Computer Science · Engineering · Mathematics · #05C50 #05C65 #15A18 #15A42 #15A60 #15A69 #Advanced Optimization Algorithms Research #Combinatorics (math.CO) #FOS: Mathematics #Matrix Theory and Algorithms #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1605.08665
openalex publication_date 2016/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The spectral p-norm of r-matrices generalizes the spectral 2-norm of 2-matrices. In 1911 Schur gave an upper bound on the spectral 2-norm of 2-matrices, which was extended in 1934 by Hardy, Littlewood, and Polya to r-matrices. Recently, Kolotilina, and independently the author, strengthened Schur's bound for 2-matrices. The main result of this paper extends the latter result to r-matrices, thereby improving the result of Hardy, Littlewood, and Polya. The proof is based on combinatorial concepts like r-partite r-matrix and symmetrant of a matrix, which appear to be instrumental in the study of the spectral p-norm in general. Thus, another application shows that the spectral p-norm and the p-spectral radius of a symmetric nonnegative r-matrix are equal whenever p≥ r. This result contributes to a classical area of analysis, initiated by Mazur and Orlicz around 1930. Additionally, a number of bounds are given on the p-spectral radius and the spectral p-norm of r-matrices and r-graphs.