Dual Spaces and Bilinear Forms in Supertropical Linear Algebra
  Continuing [5], this paper investigates finer points of supertropical vector spaces, including dual bases and bilinear forms, with supertropical versions of standard classical results such as the Gram-Schmidt theorem and Cauchy-Schwarz inequality, and change of base. We also present the supertropical version of quadratic forms, and see how they correspond to symmetric supertropical bilinear forms. 
Zur Izhakian, Manfred Knebusch, Louis Rowen
math.AC math.RA