超热带代数中矩阵的伪逆及其特征多项式
On pseudo-invereses of matrices and their characteristic polynomials in supertropical algebra
摘要：
热带代数中唯一的可逆矩阵是对角矩阵、置换矩阵及其乘积。然而，伪逆$A^\nabla$，定义为$\frac{adj(A)}{det(A)}$，其中$det(A)$是矩阵$A$的热带永久式（也称为热带行列式），继承了一些经典的代数性质，并具有一些令人惊讶的新性质。定义$B$和$B'$是热带相似的，如果$B' =A^\nabla BA$，我们检查的特征（最大）多项式的热带相似矩阵以及伪逆。其他杂项结果包括一个新的证明身份的$det(AB)$和连接稳定的权力定矩阵。
英文摘要：
The only invertible matrices in tropical algebra are diagonal matrices, permutation matrices and their products. However, the pseudo-inverse $A^\nabla$, defined as $\frac{adj(A)}{det(A)}$, with $det(A)$ being the tropical permanent (also called the tropical determinant) of a matrix $A$, inherits some classical algebraic properties and has some surprising new ones. Defining $B$ and $B'$ to be tropically similar if $B' =A^\nabla BA$, we examine the characteristic (max-)polynomials of tropically similar matrices as well as those of pseudo-inverses. Other miscellaneous results include a new proof of the identity for $det(AB)$ and a connection to stabilization of the powers of definite matrices.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
