Noncommutative algebraic equations and the noncommutative eigenvalue problem

We analyze the perturbation series for the noncommutative eigenvalue proble m AX=X lambda, where lambda is an element of a noncommutative ring, A is a matrix, and X is a column vector with entries from this ring. As a corollar y, we obtain a theorem about the structure of perturbation series for Tr x( r) where x is a solution of a noncommutative algebraic equation (for r=1 th is theorem was proved by Aschieri, Brace, Morariu, and Zumino (hep-th/00032 28), and used to study the Born-Infeld Lagrangian for the gauge group U(1)( k)).