Combinatorial methods for the spectral p-norm of hypermatrices


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 Pólya 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 Pólya. 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 back in 1930. Additionally, a number of bounds are given on the p-spectral radius and the spectral p-norm of r-matrices and r-graphs.

Publication Title

Linear Algebra and Its Applications