RT Journal Article
T1 Matrices commuting with a given normal tropical matrix
A1 Linde, J.
A1 Puente Muñoz, María Jesús de la
AB Consider the space Mnnor of square normal matrices X=(xij) over R∪{-∞}, i.e., -∞≤xij≤0 and ;bsupesup&=0. Endow Mnnor with the tropical sum ⊕ and multiplication. Fix a real matrix AεMnnor and consider the set Ω(A) of matrices in Mnnor which commute with A. We prove that Ω(A) is a finite union of alcoved polytopes; in particular, Ω(A) is a finite union of convex sets. The set ;bsupA;esup&(A) of X such that AX=XA=A is also a finite union of alcoved polytopes. The same is true for the set ′(A) of X such that AX=XA=X. A topology is given to Mnnor. Then, the set ΩA(A) is a neighborhood of the identity matrix I. If A is strictly normal, then Ω′(A) is a neighborhood of the zero matrix. In one case, Ω(A) is a neighborhood of A. We give an upper bound for the dimension of Ω′(A). We explore the relationship between the polyhedral complexes span A, span X and span(AX), when A and X commute. Two matrices, denoted A and A¯, arise from A, in connection with Ω(A). The geometric meaning of them is given in detail, for one example. We produce examples of matrices which commute, in any dimension.
PB Elsevier Science
SN 0024-3795
YR 2015
FD 2015-10
LK https://hdl.handle.net/20.500.14352/24098
UL https://hdl.handle.net/20.500.14352/24098
LA eng
DS Docta Complutense
RD 29 abr 2024