🌴Tropical Geometry Unit 4 – Tropical Linear Algebra and Matrices

Key Concepts and Definitions

• Tropical semiring consists of the real numbers together with negative infinity, with the operations of addition and maximum
• Tropical addition defined as the maximum of two numbers, denoted by $\oplus$
• Tropical multiplication defined as the usual addition of real numbers, denoted by $\otimes$
• Tropical division corresponds to subtraction in the usual sense, as the inverse operation of tropical multiplication
  • $a \oslash b = a \otimes (-b) = a - b$
• Tropical powers computed by multiplying the exponent with the base using tropical multiplication
  • $a^{\otimes n} = na$ for $n \in \mathbb{N}$
• Idempotency property states that $a \oplus a = a$ for any element $a$ in the tropical semiring
• Absorption law holds in the tropical semiring, where $a \otimes (a \oplus b) = a$ for any elements $a$ and $b$

Tropical Algebra Basics

• Tropical addition is commutative, associative, and idempotent
  • $a \oplus b = b \oplus a$
  • $(a \oplus b) \oplus c = a \oplus (b \oplus c)$
  • $a \oplus a = a$
• Tropical multiplication is commutative, associative, and distributes over tropical addition
  • $a \otimes b = b \otimes a$
  • $(a \otimes b) \otimes c = a \otimes (b \otimes c)$
  • $a \otimes (b \oplus c) = (a \otimes b) \oplus (a \otimes c)$
• Neutral elements exist for both tropical addition and multiplication
  • $a \oplus -\infty = a$ (neutral element for addition)
  • $a \otimes 0 = a$ (neutral element for multiplication)
• No additive inverses in the tropical semiring, unlike in a field
• Tropical division is not always possible, as subtraction is not always defined in the tropical semiring

Matrix Operations in the Tropical Setting

• Tropical matrix addition performed element-wise using tropical addition ($\oplus$)
• Tropical matrix multiplication involves tropical addition and multiplication, similar to the usual matrix multiplication
  • $C_{ij} = \bigoplus_{k} (A_{ik} \otimes B_{kj})$
• Tropical matrix powers computed by repeated tropical matrix multiplication
  • $A^{\otimes n} = \underbrace{A \otimes A \otimes \cdots \otimes A}_{n \text{ times}}$
• Tropical identity matrix has 0 on the diagonal and $-\infty$ elsewhere
• Tropical diagonal matrices have finite entries on the diagonal and $-\infty$ elsewhere
• Tropical triangular matrices have finite entries on and below (or above) the diagonal and $-\infty$ elsewhere
• Tropical permutation matrices obtained by permuting the rows or columns of the tropical identity matrix

Eigenvalues and Eigenvectors in Tropical Linear Algebra

• Tropical eigenvalue $\lambda$ and eigenvector $v$ of a square matrix $A$ satisfy $A \otimes v = \lambda \otimes v$
• Tropical eigenvalues can be computed using the maximum cycle mean of the weighted digraph associated with the matrix
  • Maximum cycle mean is the maximum average weight of all cycles in the graph
• Tropical eigenvectors correspond to the paths in the weighted digraph that attain the maximum cycle mean
• Existence of tropical eigenvalues and eigenvectors guaranteed for irreducible matrices (strongly connected digraphs)
• Tropical spectral theory studies the properties of tropical eigenvalues and eigenvectors
• Power method can be adapted to the tropical setting to compute the maximum cycle mean and corresponding eigenvectors

Applications in Optimization and Scheduling

• Tropical linear algebra used to model and solve optimization problems, particularly in scheduling and discrete event systems
• Shortest path problems can be formulated using tropical matrix powers
  • Entries of $A^{\otimes n}$ give the shortest path lengths between nodes in a graph after $n$ steps
• Project scheduling problems modeled using tropical algebra
  • Critical path corresponds to the tropical eigenvalue and eigenvectors of the project matrix
• Tropical optimization techniques applied in machine scheduling, railway scheduling, and synchronization problems
• Max-plus algebra, a variant of tropical algebra, widely used in modeling discrete event systems and manufacturing processes

Tropical Geometry Connections

• Tropical geometry studies geometric objects defined by tropical polynomial equations
• Tropical polynomials obtained by replacing addition and multiplication in classical polynomials with tropical operations
• Tropical hypersurfaces are the analogues of algebraic varieties in tropical geometry
  • Defined as the set of points where a tropical polynomial attains its maximum at least twice
• Tropical convexity plays a crucial role in understanding the structure of tropical hypersurfaces
• Tropical lines, tropical curves, and tropical surfaces are key objects in tropical geometry
• Connections between tropical linear spaces and classical linear spaces through the process of tropicalization
• Tropical Grassmannians and tropical flag varieties generalize the concepts of Grassmannians and flag varieties to the tropical setting

Problem-Solving Techniques

• Identify the appropriate tropical operations and structures for the given problem
• Represent the problem using tropical matrices, polynomials, or weighted digraphs
• Utilize the properties of tropical algebra, such as idempotency and the absorption law, to simplify expressions
• Apply tropical matrix operations, such as addition, multiplication, and powers, to solve linear equations and optimization problems
• Interpret the results in the context of the original problem, considering the meaning of tropical eigenvalues and eigenvectors
• Visualize tropical geometric objects using polyhedral complexes or subdivisions of Euclidean space
• Exploit the connections between tropical and classical mathematics to gain insights and solve problems

Further Reading and Resources

• "Introduction to Tropical Geometry" by Diane Maclagan and Bernd Sturmfels
  • Comprehensive textbook covering the foundations of tropical geometry and its applications
• "Tropical Mathematics" by Imre Simon
  • Seminal paper introducing the tropical semiring and its algebraic properties
• "Tropical Convexity" by Marianne Akian, Stéphane Gaubert, and Alexander Guterman
  • Survey article on the theory of tropical convexity and its connections to optimization
• "Tropical Algebraic Geometry" by Ilia Itenberg, Grigory Mikhalkin, and Eugenii Shustin
  • Monograph exploring the interplay between tropical geometry and algebraic geometry
• "Max-Plus Algebra" by Bernd Heidergott, Geert Jan Olsder, and Jacob van der Woude
  • Book focusing on the applications of max-plus algebra in discrete event systems and optimization
• "Tropical Semirings" by Zur Izhakian and Louis Rowen
  • Paper introducing the algebraic structure of tropical semirings and their properties
• Online resources, such as the "Tropical Geometry" website by Diane Maclagan and the "Tropical Geometry and Mirror Symmetry" course notes by Erwan Brugallé