Supertropical linear algebra

Supertropical algebra

We develop the algebraic polynomial theory for " supertropical algebra, " as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of ghost elements, which play the key role in our structure theory. Here, we work in a slightly more general situation over an arbitrary semiring, and develop a theory which contains the analogs of the basic theo...

The Algebra of Linear Algebra

3. You are already familiar with the concept of vector spaces in the form of …nite dimensional Euclidean vector spaces, and most of what we do in these lectures will involve these spaces (and, indeed, will also assume that the dot product operation is well-de…ned so that we can de…ne the matrix operations associated with linear transformations). However, it will also be useful to consider some ...

Supertropical Matrix Algebra Ii: Solving Tropical Equations Zur Izhakian and Louis Rowen

We continue the study of matrices over a supertropical algebra, proving the existence of a tangible adjoint of A, which provides the unique right (resp. left) quasi-inverse maximal with respect to the right (resp. left) quasi-identity matrix corresponding to A; this provides a unique maximal (tangible) solution to supertropical vector equations, via a version of Cramer’s rule. We also describe ...

Linear Algebra . A publication of the International Linear Algebra Society

Nonnegative nilpotent lower triangular completions of a nonnegative nilpotent matrix are studied. It is shown that for every natural number between the index of the matrix and its order, there exists a completion that has this number as its index. A similar result is obtained for the rank. However, unlike the case of complex completions of complex matrices, it is proved that for every nonincrea...

Trigonometric Identities, Linear Algebra, and Computer Algebra