### 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 |
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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 |

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## Cite this

Izhakian, Z., & Rowen, L. (2011). Supertropical matrix algebra.

*Israel Journal of Mathematics*,*182*(1), 383-424. https://doi.org/10.1007/s11856-011-0036-2