标题： 对称热带代数和赋值有序域上的正定矩阵的谱性质 显示英文标题

标题： Spectral Properties of Positive Definite Matrices over Symmetrized Tropical Algebras and Valued Ordered fields

摘要： 我们研究在对称化热带代数框架内正定和半正定对称矩阵的性质，这是适应有序赋值域的热带代数的扩展。 我们关注这些矩阵的特征值和特征向量。 我们证明在热带对称化设置中，正（半）定矩阵的特征值与其对角线元素一致。 然后，我们表明在赋值非阿基米德有序域上的正定矩阵的特征值的赋值图像与关联矩阵在对称化热带代数中的特征值一致。 此外，在一个通用性条件下，我们通过跟踪非阿基米德赋值和符号的映射来表征特征向量的图像，表明它们与对称化代数中的热带特征向量一致。 这些结果为热带半环上矩阵的谱理论提供了新的见解，并为实正定矩阵参数族的特征值和特征向量的对数极限提供了组合公式\ae 。 显示英文摘要

摘要： We investigate the properties of positive definite and positive semi-definite symmetric matrices within the framework of symmetrized tropical algebra, an extension of tropical algebra adapted to ordered valued fields. We focus on the eigenvalues and eigenvectors of these matrices. We prove that the eigenvalues of a positive (semi)-definite matrix in the tropical symmetrized setting coincide with its diagonal entries. Then, we show that the images by the valuation of the eigenvalues of a positive definite matrix over a valued nonarchimedean ordered field coincide with the eigenvalues of an associated matrix in the symmetrized tropical algebra. Moreover, under a genericity condition, we characterize the images of the eigenvectors under the map keeping track both of the nonarchimedean valuation and sign, showing that they coincide with tropical eigenvectors in the symmetrized algebra. These results offer new insights into the spectral theory of matrices over tropical semirings, and provide combinatorial formul\ae\ for log-limits of eigenvalues and eigenvectors of parametric families of real positive definite matrices.