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LINEAR PRESERVERS OF SYMMETRIC ARCTIC RANK OVER THE BINARY BOOLEAN SEMIRING

  • Beasley, LeRoy B. ;
  • Song, Seok-Zun
  • Received : 2016.07.29
  • Published : 2017.07.01

Abstract

A Boolean rank one matrix can be factored as $\text{uv}^t$ for vectors u and v of appropriate orders. The perimeter of this Boolean rank one matrix is the number of nonzero entries in u plus the number of nonzero entries in 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

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Acknowledgement

Supported by : National Research Foundation of Korea(NRF)