# Zero-term rank preservers of integer matrices

• Volume: 26, Issue: 2, page 155-161
• ISSN: 1509-9415

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## Abstract

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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.

## How to cite

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Seok-Zun Song, and Young-Bae Jun. "Zero-term rank preservers of integer matrices." Discussiones Mathematicae - General Algebra and Applications 26.2 (2006): 155-161. <http://eudml.org/doc/276916>.

@article{Seok2006,
abstract = {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.},
author = {Seok-Zun Song, Young-Bae Jun},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {linear operator; term-rank; zero-term rank; (P,Q,B)-operator; -operator},
language = {eng},
number = {2},
pages = {155-161},
title = {Zero-term rank preservers of integer matrices},
url = {http://eudml.org/doc/276916},
volume = {26},
year = {2006},
}

TY - JOUR
AU - Seok-Zun Song
AU - Young-Bae Jun
TI - Zero-term rank preservers of integer matrices
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2006
VL - 26
IS - 2
SP - 155
EP - 161
AB - 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.
LA - eng
KW - linear operator; term-rank; zero-term rank; (P,Q,B)-operator; -operator
UR - http://eudml.org/doc/276916
ER -

## References

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1. [1] L.B. Beasley and N.J. Pullman, Term-rank, permanent and rook-polynomial preservers, Linear Algebra Appl. 90 (1987), 33-46. Zbl0617.15001
2. [2] L.B. Beasley, S.Z. Song and S.G. Lee, Zero-term rank preserver, Linear and Multilinear Algebra. 48 (2) (2000), 313-318.
3. [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. Zbl1051.15001
4. [4] R.A. Brualdi and H.J. Ryser, Combinatorial Matrix Theory, Encyclopedia of Mathematics and its Applications, Vol. 39, Cambridge University Press, Cambridge 1991.
5. [5] C R. Johnson and J.S. Maybee, Vanishing minor conditions for inverse zero patterns, Linear Algebra Appl. 178 (1993), 1-15. Zbl0767.15003
6. [6] M. Marcus, Linear operations on matrices, Amer. Math. Monthly 69 (1962), 837-847. Zbl0108.01104
7. [7] H. Minc, Permanents, Encyclopedia of Mathematics and its Applications, Vol. 6, Addison-Wesley Publishing Company, Reading, Massachusetts 1978.
8. [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.

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