  In this paper we prove a conjecture regarding the form of the Born-Infeld
Lagrangian with a U(1)^2n gauge group after the elimination of the auxiliary
fields. We show that the Lagrangian can be written as a symmetrized trace of
Lorentz invariant bilinears in the field strength. More generally we prove a
theorem regarding certain solutions of unilateral matrix equations of arbitrary
order. For solutions which have perturbative expansions in the matrix
coefficients, the solution and all its positive powers are sums of terms which
are symmetrized in all the matrix coefficients and of terms which are
commutators.
