# Rank-preserver of Matrices over Chain Semiring

• Published : 2006.03.23

#### Abstract

For a rank-1 matrix A, there is a factorization as $A=ab^t$, the product of two vectors a and b. We characterize the linear operators that preserve rank and some equivalent condition of rank-1 matrices over a chain semiring. We also obtain a linear operator T preserves the rank of rank-1 matrices if and only if it is a form (P, Q, B)-operator with appropriate permutation matrices P and Q, and a matrix B with all nonzero entries.

#### References

1. L. B. Beasley and N. J. Pullman, Boolean rank-preserving operators and Boolean rank-1 spaces, Linear Algebra Appl., 59(1984), 55-77. https://doi.org/10.1016/0024-3795(84)90158-7
2. L. B. Beasley and N. J. Pullman, Fuzzy Rank-Preserving Operators, Linear Algebra Appl., 73(1986), 197-211. https://doi.org/10.1016/0024-3795(86)90240-5
3. L. B. Beasley and N. J. Pullman, Term-rank, permanent and rook-polynomial preservers, Linear Algebra Appl., 90(1987), 33-46. https://doi.org/10.1016/0024-3795(87)90302-8
4. L. B. Beasley, D. A. Gregory and N. J. Pullman, Nonnegative Rank-Preserving Operators, Linear Algebra Appl., 65(1985), 207-223. https://doi.org/10.1016/0024-3795(85)90098-9
5. S. Z. Song, Linear operators that preserve column rank of fuzzy matrices, Fuzzy Sets and Systems, 62(1994), 311-317. https://doi.org/10.1016/0165-0114(94)90115-5
6. S. Z. Song, L. B. Beasley, G. S. Cheon and Y. B. Jun, Rank and perimeter preservers of Boolean rank-1 matrices, J. Korean Math. Soc., 41(2)(2004), 397-406. https://doi.org/10.4134/JKMS.2004.41.2.397