TY - JOUR
T1 - Supertropical linear algebra
AU - Izhakian, Zur
AU - Knebusch, Manfred
AU - Rowen, Louis
PY - 2013
Y1 - 2013
N2 - The objective of this paper is to lay out the algebraic theory of supertropical vector spaces and linear algebra, utilizing the key antisymmetric relation of "ghost surpasses". Special attention is paid to the various notions of "base", which include d-base and s-base, and these are compared to other treatments in the tropical theory. Whereas the number of elements in various d-bases may differ, it is shown that when an s-base exists, it is unique up to permutation and multiplication by scalars, and can be identified with a set of "critical" elements. Then we turn to orthogonality of vectors, which leads to supertropical bilinear forms and a supertropical version of the Gram matrix, including its connection to linear dependence. We also obtain a supertropical version of a theorem of Artin, which says that if g-orthogonality is a symmetric relation, then the underlying bilinear form is (supertropically) symmetric.
AB - The objective of this paper is to lay out the algebraic theory of supertropical vector spaces and linear algebra, utilizing the key antisymmetric relation of "ghost surpasses". Special attention is paid to the various notions of "base", which include d-base and s-base, and these are compared to other treatments in the tropical theory. Whereas the number of elements in various d-bases may differ, it is shown that when an s-base exists, it is unique up to permutation and multiplication by scalars, and can be identified with a set of "critical" elements. Then we turn to orthogonality of vectors, which leads to supertropical bilinear forms and a supertropical version of the Gram matrix, including its connection to linear dependence. We also obtain a supertropical version of a theorem of Artin, which says that if g-orthogonality is a symmetric relation, then the underlying bilinear form is (supertropically) symmetric.
KW - Change of base semirings
KW - Gram matrix.
KW - Linear algebra
KW - Linear and bilinear forms
KW - Supertropical vector spaces
KW - Tropical algebra
UR - http://www.scopus.com/inward/record.url?scp=84886875073&partnerID=8YFLogxK
U2 - 10.2140/pjm.2013.266.43
DO - 10.2140/pjm.2013.266.43
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AN - SCOPUS:84886875073
SN - 0030-8730
VL - 266
SP - 43
EP - 75
JO - Pacific Journal of Mathematics
JF - Pacific Journal of Mathematics
IS - 1
ER -