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arxiv: 0902.2159 · v3 · submitted 2009-02-12 · 🧮 math.AC · math.CO

Supertropical Matrix Algebra II: Solving tropical equations

classification 🧮 math.AC math.CO
keywords supertropicalmatrixtangibleadjointalgebraeigenvectorsequationsleft
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We continue the study of matrices over a supertropical algebra, proving the existence of a tangible adjoint of $A$, which provides the unique right (resp. left) quasi-inverse maximal with respect to the right (resp. left) quasi-identity matrix corresponding to $A$; this provides a unique maximal (tangible) solution to supertropical vector equations, via a version of Cramer's rule. We also describe various properties of this tangible adjoint, and use it to compute supertropical eigenvectors, thereby producing an example in which an $n\times n$ matrix has $n$ distinct supertropical eigenvalues but their supertropical eigenvectors are tropically dependent.

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