## Supertropical matrix algebra.(English)Zbl 1215.15018

The objective of this paper is to develop a general algebraic theory of a supertropical matrix algebra, extending previous results of the first author. The main results are as follows:
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The tropical determinant (i.e., permanent) is multiplicative when all the determinants involved are tangible.
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There exists an adjoint matrix adj$$(A)$$ such that the matrix $$A\cdot\text{adj}(A)$$ behaves much like the identity matrix (times $$|A|$$).
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Every matrix $$A$$ is a supertropical root of its Hamilton-Cayley polynomial $$f_A$$. If these roots are distinct, then $$A$$ is conjugate (in a certain supertropical sense) to a diagonal matrix.
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The tropical determinant of a matrix $$A$$ is a ghost iff the rows of $$A$$ are tropically dependent, iff the columns of $$A$$ are tropically dependent.
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Every root of $$f_A$$ is a “supertropical” eigenvalue of $$A$$ (appropriately defined), and has a tangible supertropical eigenvector.

### MSC:

 15A30 Algebraic systems of matrices 15A15 Determinants, permanents, traces, other special matrix functions 15A18 Eigenvalues, singular values, and eigenvectors

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### References:

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