Journal of the
Korean Mathematical Society JKMS

A Boolean rank one matrix can be factored as ${\bf u}{\bf v}^t$ for vectors ${\bf u}$ and ${\bf v}$ of appropriate orders. The perimeter of this Boolean rank one matrix is the number of nonzero entries in ${\bf u}$ plus the number of nonzero entries in ${\bf v}$. A Boolean matrix of Boolean rank $k$ is the sum of $k$ Boolean rank one matrices, a rank one decomposition. The perimeter of a Boolean matrix $A$ of Boolean rank $k$ is the minimum over all Boolean rank one decompositions of $A$ of the sums of perimeters of the Boolean rank one matrices. The arctic rank of a Boolean matrix is one half the perimeter. In this article we characterize the linear operators that preserve the symmetric arctic rank of symmetric Boolean matrices.

Keywords: linear operator, preserve, symmetric arctic rank, $(P,P^t)$-operator

MSC numbers: Primary 15A86, 15A04, 15B34