Dual spaces and bilinear forms in supertropical linear algebra

Abstract

Continuing 4, this article 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–Schwartz inequality, and change of base. We also present the supertropical version of quadratic forms, and see how they correspond to symmetric supertropical bilinear forms.

Keywords:

• tropical algebra
• vector space
• linear algebra
• d-base
• s-base
• dual base
• change of base semirings
• bilinear form

AMS Subject Classifications::

• Primary: 15A03
• 15A09
• 15A15
• 16Y60
• Secondary: 14T05
• 15A33
• 20M18
• 51M20

Acknowledgements

The work of the first and third authors has been supported by the Israel Science Foundation, grant 448/09. The second author was supported in part by the Gelbart Institute at Bar-Ilan University, the Minerva Foundation at Tel-Aviv University, the Mathematics Department of Bar-Ilan University, and the Emmy Noether Institute.