Strong preservers of symmetric arctic rank of nonnegative real matrices
J. Korean Math. Soc. 2019 Vol. 56, No. 6, 1503-1514
Published online November 1, 2019
LeRoy B. Beasley, Luis Hernandez Encinas, Seok-Zun Song
Utah State University; Spanish National Research Council (CSIC); Korean Institute for Advanced Study
Abstract : A rank $1$ matrix has a factorization as ${\bf u}{\bf v}^t$ for vectors ${\bf u}$ and ${\bf v}$ of some orders. The arctic rank of a rank $1$ matrix is the half number of nonzero entries in ${\bf u}$ and ${\bf v}$. A matrix of rank $k$ can be expressed as the sum of $k$ rank $1$ matrices, a rank $1$ decomposition. The arctic rank of a matrix $A$ of rank $k$ is the minimum of the sums of arctic ranks of the rank $1$ matrices over all rank $1$ decomposition of $A$. In this paper we obtain characterizations of the linear operators that strongly preserve the symmetric arctic ranks of symmetric matrices over nonnegative reals.
Keywords : linear operator, $(P,P^t, B)$-operator, weighted cell, symmetric arctic rank
MSC numbers : Primary 15A86, 15A04, 15B34
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