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

Matrix inequalities play a pivotal role in mathematics, generalizing scalar inequalities and providing insights into linear operator structures. However, the widely used Löwner 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.