Matrix Ordering through Spectral and Nilpotent Structures in Totally Ordered Complex Number Fields
Shih-Yu Chang
TL;DR
The paper develops a comprehensive framework to compare arbitrary complex matrices by combining their spectral and nilpotent structures into the Spectral and Nilpotent Ordering (SNO). It introduces a total order on complex numbers, extends majorization to complex-valued vectors and functions, and provides a detailed Jordan-block based representation $\mathfrak{R}(\bm{X})$ to formalize matrix comparisons; Loewner order emerges as a special case. The work further derives monotonicity and convexity conditions for complex-valued functions under SNO, supported by a complex Schur–Ostrowski criterion and Hansen–Pedersen style convexity results. Collectively, these contributions extend matrix analysis to non-Hermitian contexts with potential impact on operator theory, quantum information, and control theory by offering a structured way to reason about spectral and nilpotent components under functional calculus.
Abstract
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.