### Abstract

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

*Israel Journal of Mathematics*,

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

**Supertropical matrix algebra.** / Izhakian, Zur; Rowen, Louis .

Research output: Contribution to journal › Article

*Israel Journal of Mathematics*, vol. 182, no. 1, pp. 383-424. https://doi.org/10.1007/s11856-011-0036-2

}

TY - JOUR

T1 - Supertropical matrix algebra

AU - Izhakian, Zur

AU - Rowen, Louis

PY - 2011/3

Y1 - 2011/3

N2 - 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.

AB - 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.

U2 - 10.1007/s11856-011-0036-2

DO - 10.1007/s11856-011-0036-2

M3 - Article

VL - 182

SP - 383

EP - 424

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -