Combinatorial methods for the spectral p-norm of hypermatrices
نویسنده
چکیده
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 new 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.
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