AI 中文总结
该研究探究由数域参数化的整数矩阵的同谱性，利用特征多项式判别式与Krylov子空间性质推导谱确定性条件，开发算法并公开实现，用于在小数域上寻找所有同谱伴侣。
AI 中文摘要
我们研究由代数数域参数化的整数矩阵的同谱性概念。给定一个数域和一个对称整数矩阵，我们探究用该域中元素构成的正交矩阵对该整数矩阵进行共轭时，何时能生成新的整数矩阵。我们的结果涉及相关谱确定性概念的充分条件，并给出当这些条件不适用时正交矩阵的约束。该结果使用了特征多项式的判别式和Krylov子空间的性质。我们利用该理论开发了一种算法，用于在给定的（小）数域上寻找所有同谱伴侣，该算法的实现版本已公开提供。
英文摘要
We investigate a notion of cospectrality for integer matrices that is parameterized by algebraic number fields. Given a number field and a symmetric integer matrix, we wonder when conjugating the integer matrix by an orthogonal matrix with entries in the given field can produce new integer matrices. Our results concern sufficient conditions for the associated notion of spectral determination, and we give constraints on the orthogonal matrices when the conditions are not applicable. The results use the discriminant of the characteristic polynomial and properties of Krylov subspaces. We leverage the theory to develop an algorithm to find all cospectral mates over a given (small) field. An implementation of the algorithm is made available.
Comments35 pages, 7 tables, 1 figure
