2025/01/17 by Shih Yu Chang, Chang, Shih-Yu · 2 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #FOS: Physical sciences #Finite Group Theory Research #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.2501.10603
openalex publication_date 2025/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.