The authors show results for commuting complex matrices over two semirings: \roster \item"(i)" {max algebra: $(R_+, \max, \times)$}, \item"(ii)" {non-negative linear algebra: $(R_+, +, \times)$}, \endroster where $R_+$ is the set of non-negative real numbers. They illustrate several numerical examples over the semiring $(R\cup \{-\infty\}, \max, +)$, which is isomorphic to the max algebra. They adopt the notation $\lambda(A)$ for the max algebraic Perron root of $A\in\Bbb R_{+}^{n\times n}$ and call the set of max eigenvectors of $A$ associated with a max eigenvalue an eigencone of $A$. \par They investigate the existence of a common eigenvector of a set of non-negative matrices. In particular, they prove that the intersection of eigencones of pairwise commuting matrices contains a nonzero vector. They analyse the eigenvalues of polynomials of commuting matrices $A_1, \dots , A_r\in\Bbb R_{+}^{n\times n}$ and present inequalities for $\lambda(A_i)$, $i=1, \dots, r$. They state results on the intersection of the principal eigencones of commuting matrices and their relationship with the spectral projectors. \par The authors are concerned with the digraph of $A\in\Bbb R_{+}^{n\times n}$, the Frobenius normal form and max algebraic spectral theory. For commuting matrices, they characterise the Perron root of certain classes in the reduced digraph. Moreover, they analyse the Frobenius normal form of commuting matrices where all classes of $A_i$, $i=1, \dots, r$, have distinct Perron roots. They observe some equivalent conditions on the coincidence of the transitive closures and spectral classes of the associated reduced digraphs for this particular case. Note that most of their results are also true in the non-negative linear algebra. \par Specifically, the authors focus on the intersection of eigencones of commuting matrices in the max algebra. Inspired by the common eigenvector scaling and known results on the Boolean algebra, they prove that the critical digraphs of two commuting irreducible matrices have a common eigennode.