Supertropical matrix algebra

By Zur Izhakian, Louis H. Rowen. 2008

81 citations recorded by the source corpus.

AbstractThe objective of this paper is to develop a general algebraic theory of supertropical matrix algebra, extending [11]. Our main results are as follows: The tropical determinant (i.e., permanent) is multiplicative when all the determinants involved are tangible.There exists an adjoint matrix adj(A) such that the matrix A adj(A) behaves much like the identity matrix (times |A|).Every matrix A is a supertropical root of its Hamilton-Cayley polynomial fA. If these roots are distinct, then A is conjugate (in a certain supertropical sense) to a diagonal matrix.The tropical determinant of a matrix A is a ghost iff the rows of A are tropically dependent, iff the columns of A are tropically dependent.Every root of fA is a “supertropical” eigenvalue of A (appropriately defined), and has a tangible supertropical eigenvector.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cc4014a611790b8effabd83253afe393f52c6817

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Louis H. Rowen
2. Zur Izhakian
3. Manfred Knebusch
4. Marianne Akian
5. Stephane Gaubert
6. Alexander E. Guterman

In Computational Theory and Mathematics

1. Louis H. Rowen
2. Zur Izhakian
3. Manfred Knebusch
4. Marianne Akian
5. Stephane Gaubert
6. Alexander E. Guterman

In Geometry and Topology

1. Zur Izhakian
2. Louis H. Rowen
3. Manfred Knebusch

Citations in the corpus

• Supertropical matrix algebra II: Solving tropical equations (2009, methodology): “Here are some results from [6], which are reproved as easy applications of Theorem 2.”
• Supertropical semirings and supervaluations (2010, background): “ us to specialize to the max-plus theory. On the other hand, the ghost surpassing relation appears in virtually every supertropical theorem proved so far, especially in supertropical matrix theory in [IR2] and [IR3]. In the present paper the ghost surpassing relation is the essential gadget for unde”
• The Tropical Rank of a Tropical Matrix (2006, methodology): “The main goal of this article is a further development of the basics of tropical matrix algebra over the extended tropical semiring, (𝕋, ⊕, ⊙), as presented in [ 5 ]; we also use some of the terminology used in [ 8 , 9 ].”
• Supertropical linear algebra (2010, methodology): “• |adj(A)| = |A|n−1 [Izhakian and Rowen 2011a].”