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Combinatorial methods for the spectral p-norm of hypermatrices
The spectral -norm of -matrices generalizes the spectral -norm of
-matrices. In 1911 Schur gave an upper bound on the spectral -norm of
-matrices, which was extended in 1934 by Hardy, Littlewood, and Polya to
-matrices. Recently, Kolotilina, and independently the author, strengthened
Schur's bound for -matrices. The main result of this paper extends the
latter result to -matrices, thereby improving the result of Hardy,
Littlewood, and Polya.
The proof is based on combinatorial concepts like -partite -matrix and
symmetrant of a matrix, which appear to be instrumental in the study of the
spectral -norm in general. Thus, another application shows that the spectral
-norm and the -spectral radius of a symmetric nonnegative -matrix are
equal whenever . 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 -spectral radius and the
spectral -norm of -matrices and -graphs.Comment: 29 pages. Credit has been given to Ragnarsson and Van Loan for the
symmetrant of a matri
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