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.

翻译：我们在对称化热带代数（一种适用于有序赋值域的热带代数扩展）的框架内研究正定与半正定对称矩阵的性质。我们重点关注这些矩阵的特征值与特征向量。我们证明，在热带对称化设定下，正定（半正定）矩阵的特征值与其对角元一致。随后，我们证明在非阿基米德赋值有序域上，一个正定矩阵的特征值经赋值映射后的像，与对称化热带代数中一个关联矩阵的特征值一致。此外，在一个一般性条件下，我们刻画了特征向量在同时记录非阿基米德赋值与符号的映射下的像，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与特征向量之对数极限给出了组合公式。