Abstract
The 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 f A . 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 f A is a “supertropical” eigenvalue of A (appropriately defined), and has a tangible supertropical eigenvector.
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 f A . 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 f A is a “supertropical” eigenvalue of A (appropriately defined), and has a tangible supertropical eigenvector.
| Original language | English |
|---|---|
| Pages (from-to) | 383-424 |
| Number of pages | 42 |
| Journal | Israel Journal of Mathematics |
| Volume | 182 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - Mar 2011 |