Dependence of Supertropical Eigenspaces

We study the pathology that causes tropical eigenspaces of distinct supertropical eigenvalues of a nonsingular matrix $A$, to be dependent. We show that in lower dimensions the eigenvectors of distinct eigenvalues are independent, as desired. The index set that differentiates between subsequent essential monomials of the characteristic polynomial, yields an eigenvalue $\lambda$, and corresponds to the columns of the eigenmatrix $A+\lambda I$ from which the eigenvectors are taken. We ascertain the cause for failure in higher dimensions, and prove that independence of the eigenvectors is recovered in case a certain "difference criterion" holds, defined in terms of disjoint differences between index sets of subsequent coefficients. We conclude by considering the eigenvectors of the matrix A^\nabla : = \det(A)^{-1}\adj(A) and the connection of the independence question to generalized eigenvectors.Comment: The first author is sported by the French Chateaubriand grant and INRIA postdoctoral fellowshi

On the coefficients of the max-algebraic characteristic polynomial and equation

summary:No polynomial algorithms are known for finding the coefficients of the characteristic polynomial and characteristic equation of a matrix in max- algebra. The following are proved: (1) The task of finding the max-algebraic characteristic polynomial for permutation matrices encoded using the lengths of their constituent cycles is NP-complete. (2) The task of finding the lowest order finite term of the max-algebraic characteristic polynomial for a $\lbrace 0,-\infty \rbrace$ matrix can be converted to the assignment problem. (3) The task of finding the max-algebraic characteristic equation of a $\lbrace 0,-\infty \rbrace$ matrix can be converted to that of finding the conventional characteristic equation for a $\lbrace 0,1\rbrace$ matrix and thus it is solvable in polynomial time