Journal of the
Korean Mathematical Society JKMS

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

Supported by: This work was supported under the framework of international cooperation program managed by the National Research Foundation of Korea (2017K2A9A1A01092970, FY2017) and this research was supported by the 2019 scientific promotion program funded by Jeju National University.