标题： 通过全序复数域中的谱结构和幂零结构进行矩阵排序 显示英文标题

标题： Matrix Ordering through Spectral and Nilpotent Structures in Totally Ordered Complex Number Fields

摘要： 矩阵不等式在数学中起着关键作用，它推广了标量不等式，并提供了对线性算子结构的见解。 然而，广泛使用的Löwner序，依赖于实数特征值，仅适用于厄米特矩阵，这限制了其在非厄米特系统中的适用性，而非厄米特系统在非厄米特物理等领域越来越相关。 为克服这一问题，我们开发了一种复数的全序关系，使得可以比较具有复数特征值的一般矩阵的谱分量。 在此基础上，我们引入了谱和幂零序(SNO)，这是一种针对相同维度任意矩阵的部分序。 我们进一步建立了复值函数的主要序理论框架，这有助于精炼SNO并分析谱分量。 另一个结果是将Schur--Ostrowski准则扩展到复数域。 此外，我们使用一种广义的支配序来表征矩阵函数的若尔当块，这有助于对不可对角化矩阵进行系统分析。 最后，我们在SNO框架下推导了函数的单调性和凸性条件，为推进矩阵分析奠定了新的数学基础。 显示英文摘要

摘要： Matrix inequalities play a pivotal role in mathematics, generalizing scalar inequalities and providing insights into linear operator structures. However, the widely used L\"owner ordering, which relies on real-valued eigenvalues, is limited to Hermitian matrices, restricting its applicability to non-Hermitian systems increasingly relevant in fields like non-Hermitian physics. To overcome this, we develop a total ordering relation for complex numbers, enabling comparisons of the spectral components of general matrices with complex eigenvalues. Building on this, we introduce the Spectral and Nilpotent Ordering (SNO), a partial order for arbitrary matrices of the same dimensions. We further establish a theoretical framework for majorization ordering with complex-valued functions, which aids in refining SNO and analyzing spectral components. An additional result is the extension of the Schur--Ostrowski criterion to the complex domain. Moreover, we characterize Jordan blocks of matrix functions using a generalized dominance order for nilpotent components, facilitating systematic analysis of non-diagonalizable matrices. Finally, we derive monotonicity and convexity conditions for functions under the SNO framework, laying a new mathematical foundation for advancing matrix analysis.