A Glance at Tropical Operations and Tropical Linear Algebra

The tropical semiring is ℝ ∪ {∞} with the operations x ⊕ y = min{x, y}, x ⊕ ∞ = ∞ ⊕ x = x, x ⊙ y = x + y, x ⊙ ∞ = ∞ ⊙ y = ∞. This paper explores how ideas from classical algebra and linear algebra over the real numbers such as polynomials, roots of polynomials, lines, matrices and matrix operations, determinants, eigen values and eigen vectors would appear in tropical mathematics. It uses numerous computed examples to illustrate these concepts and explores the relationship between certain tropical matrices and graph theory, using this to provide proofs of some tropical computations