Tropical linear algebra: orbits and the core of a matrix
We present a selection of classical and new results in tropical
linear algebra with emphasis on combinatorial aspects. First we
discuss tropicalisation of fundamental linear-algebraic problems
such as linear equations, eigenproblem and characteristic polynomial.
This will be followed by the question of reachability of eigenspaces
by matrix orbits. The core of a matrix is the intersection of the
column spaces of all powers of this matrix. We present a tropical
version of Pullman's theorem characterising the conventional core
as the Minkowski sum of all eigenspaces of all powers. The proof
is universal: it covers both the conventional and tropical case,
yielding new results also on periodicity of eigenspaces of matrix
powers. (Joint work with Hans Schneider and Sergei Sergeev.)