Rank and perimeter preserver of rank-1 matrices over max algebra

Seok-Zun Song; Kyung-Tae Kang

Discussiones Mathematicae - General Algebra and Applications (2003)

  • Volume: 23, Issue: 2, page 125-137
  • ISSN: 1509-9415

Abstract

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For a rank-1 matrix A = a ⊗ b t over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T(A) = U ⊗ A ⊗ V, or T ( A ) = U ⊗ A t ⊗ V with some monomial matrices U and V.

How to cite

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Seok-Zun Song, and Kyung-Tae Kang. "Rank and perimeter preserver of rank-1 matrices over max algebra." Discussiones Mathematicae - General Algebra and Applications 23.2 (2003): 125-137. <http://eudml.org/doc/287656>.

@article{Seok2003,
abstract = {For a rank-1 matrix $A = a ⊗ b^\{t\}$ over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T(A) = U ⊗ A ⊗ V, or $T(A) = U ⊗ A^\{t\} ⊗ V$ with some monomial matrices U and V.},
author = {Seok-Zun Song, Kyung-Tae Kang},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {max algebra; semiring; linear operator; monomial; rank; dominate; perimeter; (U,V)-operator; linear preserver},
language = {eng},
number = {2},
pages = {125-137},
title = {Rank and perimeter preserver of rank-1 matrices over max algebra},
url = {http://eudml.org/doc/287656},
volume = {23},
year = {2003},
}

TY - JOUR
AU - Seok-Zun Song
AU - Kyung-Tae Kang
TI - Rank and perimeter preserver of rank-1 matrices over max algebra
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2003
VL - 23
IS - 2
SP - 125
EP - 137
AB - For a rank-1 matrix $A = a ⊗ b^{t}$ over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T(A) = U ⊗ A ⊗ V, or $T(A) = U ⊗ A^{t} ⊗ V$ with some monomial matrices U and V.
LA - eng
KW - max algebra; semiring; linear operator; monomial; rank; dominate; perimeter; (U,V)-operator; linear preserver
UR - http://eudml.org/doc/287656
ER -

References

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  1. [1] R.B. Bapat, A max version of the Perron-Frebenius theorem, Linear Algebra Appl. 275-276 (1998), 3-18. 
  2. [2] R.B. Bapat, S. Pati and S.-Z. Song, Rank preservers of matrices over max algebra, Linear and Multilinear Algebra 48 (2000), 149-164. Zbl0971.15006
  3. [3] L.B. Beasley and N.J. Pullman, Boolean rank-preserving operators and Boolean rank-1 spaces, Linear Algebra Appl. 59 (1984), 55-77. Zbl0536.20044
  4. [4] L.B. Beasley, S.-Z. Song and S.-G. Lee, Zero term rank preservers, Linear and Multilinear Algebra 48 (2001), 313-318. Zbl0987.15002
  5. [5] S.-Z. Song and S.-R. Park, Maximal column rank preservers of fuzzy matrices, Discuss. Math. - Gen. Algebra Appl. 21 (2001), 207-218. Zbl1006.15018

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