## Supertropical matrix algebra. II: Solving tropical equations.(English)Zbl 1277.15013

Summary: 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.
For part I, cf. [Isr. J. Math. 182, 383–424 (2011; Zbl 1215.15018)].

### MSC:

 15A30 Algebraic systems of matrices 15A15 Determinants, permanents, traces, other special matrix functions 15A18 Eigenvalues, singular values, and eigenvectors 15A80 Max-plus and related algebras 14T05 Tropical geometry (MSC2010) 16S50 Endomorphism rings; matrix rings 16Y60 Semirings

Zbl 1215.15018
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### References:

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