On a proof of the general version of the spectral theorem in max-plus algebra

The so-called spectral theorem for square irreducibe matrices is well-known in the max-plus community. The theorem is a fundamental result concerning matrix powers and eigenvalues in the context of max-plus algebra and forms the basis for many results. This paper aims at giving a complete proof of the above spectral property in its full generality. In particular, the distinction will be highlighted between the graph cyclicity of a matrix (the cyclicity of the graph of the matrix) and the cyclicity of the matrix itself.