• Beasley, LeRoy B. (Department of Mathematics and Statistics Utah State University) ;
  • Encinas, Luis Hernandez (Institute of Physical and Information Technologies Spanish National Research Council (CSIC)) ;
  • Song, Seok-Zun (Department of Mathematics Jeju National University)
  • Received : 2018.11.11
  • Accepted : 2019.02.07
  • Published : 2019.11.01


A rank 1 matrix has a factorization as $uv^t$ for vectors u and v of some orders. The arctic rank of a rank 1 matrix is the half number of nonzero entries in u and 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.


Supported by : National Research Foundation of Korea


  1. L. B. Beasley, A. E. Guterman, and Y. Shitov, The arctic rank of a Boolean matrix, J. Algebra 433 (2015), 168-182.
  2. L. B. Beasley and N. J. Pullman, Boolean-rank-preserving operators and Boolean-rank-1 spaces, Linear Algebra Appl. 59 (1984), 55-77.
  3. L. B. Beasley and N. J. Pullman, Term-rank, permanent, and rook-polynomial preservers, Linear Algebra Appl. 90 (1987), 33-46.
  4. L. B. Beasley and S.-Z. Song, Primitive symmetric matrices and their preservers, Linear Multilinear Algebra 65 (2017), no. 1, 129-139.
  5. L. B. Beasley and S.-Z. Song, Symmetric arctic ranks of nonnegative matrices and their linear preservers, Linear Multilinear Algebra 65 (2017), no. 10, 2000-2010.
  6. K. H. Kim and F. W. Roush, Kapranov rank vs. tropical rank, Proc. Amer. Math. Soc. 134 (2006), no. 9, 2487-2494.
  7. T. Markham, Factorizations of completely positive matrices, Proc. Cambridge Philos. Soc. 69 (1971), 53-58.
  8. S. Pierce, Algebraic sets, polynomials, and other functions, Linear and Multilinear Algebra 33 (1992), no. 1-2, 31-52.
  9. S. Z. Song, L. B. Beasley, P. Mohindru, and R. Pereira, Preservers of completely positive matrix rank, Linear Multilinear Algebra 64 (2016), no. 7, 1258-1265.
  10. S.-Z. Song, K.-T. Kang, and L. B. Beasley, Linear operators that preserve perimeters of matrices over semirings, J. Korean Math. Soc. 46 (2009), no. 1, 113-123.