The zero-term rank of a matrix is the minimum number of lines (row or columns) needed to cover all the zero entries of the given matrix. We characterize the linear operators that preserve the zero-term rank of the m × n integer matrices. That is, a linear operator T preserves the zero-term rank if and only if it has the form T(A)=P(A ∘ B)Q, where P, Q are permutation matrices and A ∘ B is the Schur product with B whose entries are all nonzero integers.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1109,
author = {Seok-Zun Song and Young-Bae Jun},
title = {Zero-term rank preservers of integer matrices},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {26},
year = {2006},
pages = {155-161},
zbl = {1131.15003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1109}
}
Seok-Zun Song; Young-Bae Jun. Zero-term rank preservers of integer matrices. Discussiones Mathematicae - General Algebra and Applications, Tome 26 (2006) pp. 155-161. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1109/
[000] [1] L.B. Beasley and N.J. Pullman, Term-rank, permanent and rook-polynomial preservers, Linear Algebra Appl. 90 (1987), 33-46. | Zbl 0617.15001
[001] [2] L.B. Beasley, S.Z. Song and S.G. Lee, Zero-term rank preserver, Linear and Multilinear Algebra. 48 (2) (2000), 313-318.
[002] [3] L.B. Beasley, Y.B. Jun and S.Z. Song, Zero-term ranks of real matrices and their preserver, Czechoslovak Math. J. 54 (129) (2004), 183-188. | Zbl 1051.15001
[003] [4] R.A. Brualdi and H.J. Ryser, Combinatorial Matrix Theory, Encyclopedia of Mathematics and its Applications, Vol. 39, Cambridge University Press, Cambridge 1991.
[004] [5] C R. Johnson and J.S. Maybee, Vanishing minor conditions for inverse zero patterns, Linear Algebra Appl. 178 (1993), 1-15. | Zbl 0767.15003
[005] [6] M. Marcus, Linear operations on matrices, Amer. Math. Monthly 69 (1962), 837-847. | Zbl 0108.01104
[006] [7] H. Minc, Permanents, Encyclopedia of Mathematics and its Applications, Vol. 6, Addison-Wesley Publishing Company, Reading, Massachusetts 1978.
[007] [8] C.K. Li and N.K. Tsing, Linear preserver problems: A brief introduction and some special techniques, Linear Algebra Appl. 162-164 (1992), 217-235.