On the characteristic polynomial of a supertropical adjoint matrix

Abstract

Let χ(A) denote the characteristic polynomial of a matrix A over a field; a standard result of linear algebra states that χ(A −1 ) is the reciprocal polynomial of χ(A). More formally, the condition χ n (X)χ k (X −1 ) = χ n−k (X) holds for any invertible n × n matrix X over a field, where χ i (X) denotes the coefficient of λ n−i in the characteristic polynomial det(λI − X). We confirm a recent conjecture of Niv by proving the tropical analogue of this result.

Tropical compound matrix identities

“…We then establish the connection to their corresponding eigenvalues, and deal with the special case of equality in Corollaries 6.6 and 6.5 respectively. Note that the supertropical version of the identities in (1.1) were proved in [Niv14a], [Niv15] and [Shi16].…”